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C H A P T E R

2B(t)

N(t)

T(t)

DeterminantsWith each square matrix, it is possible to associate a real number called the determinantof the matrix. The value of this number will tell us whether the matrix is singular.

In Section 1, the definition of the determinant of a matrix is given. In Section 2,we study properties of determinants and derive an elimination method for evaluatingdeterminants. The elimination method is generally the simplest method to use forevaluating the determinant of an n × n matrix when n > 3. In Section 3, we seehow determinants can be applied to solving n × n linear systems and how they can beused to calculate the inverse of a matrix. Applications of determinants to cryptographyand to Newtonian mechanics are also presented in Section 3. Further applications ofdeterminants are presented in Chapters 3 and 6.

2.1 The Determinant of a Matrix

With each n × n matrix A, it is possible to associate a scalar, det(A), whose value willtell us whether the matrix is nonsingular. Before proceeding to the general definition,let us consider the following cases:

Case 1. 1 × 1 Matrices If A = (a) is a 1×1 matrix, then A will have a multiplicativeinverse if and only if a �= 0. Thus, if we define

det(A) = a

then A will be nonsingular if and only if det(A) �= 0.

Case 2. 2 × 2 Matrices Let

A =⎧⎪⎩a11 a12

a21 a22

⎫⎪⎭By Theorem 1.5.2, A will be nonsingular if and only if it is row equivalent to I . Then,if a11 �= 0, we can test whether A is row equivalent to I by performing the followingoperations:

84

2.1 The Determinant of a Matrix 85

1. Multiply the second row of A by a11⎧⎪⎩ a11 a12

a11a21 a11a22

⎫⎪⎭2. Subtract a21 times the first row from the new second row⎧⎪⎩a11 a12

0 a11a22 − a21a12

⎫⎪⎭Since a11 �= 0, the resulting matrix will be row equivalent to I if and only if

a11a22 − a21a12 �= 0 (1)

If a11 = 0, we can switch the two rows of A. The resulting matrix⎧⎪⎩a21 a22

0 a12

⎫⎪⎭will be row equivalent to I if and only if a21a12 �= 0. This requirement is equivalent tocondition (1) when a11 = 0. Thus, if A is any 2 × 2 matrix and we define

det(A) = a11a22 − a12a21

then A is nonsingular if and only if det(A) �= 0.

Notation

We can refer to the determinant of a specific matrix by enclosing the array betweenvertical lines. For example, if

A =⎧⎪⎩3 4

2 1

⎫⎪⎭then ∣∣∣∣3 4

2 1

∣∣∣∣represents the determinant of A.

Case 3. 3 × 3 Matrices We can test whether a 3×3 matrix is nonsingular by perform-ing row operations to see if the matrix is row equivalent to the identity matrix I . Tocarry out the elimination in the first column of an arbitrary 3 × 3 matrix A, let us firstassume that a11 �= 0. The elimination can then be performed by subtracting a21/a11

times the first row from the second and a31/a11 times the first row from the third:

⎧⎪⎪⎪⎪⎪⎩a11 a12 a13

a21 a22 a23

a31 a32 a33

⎫⎪⎪⎪⎪⎪⎭ →

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

a11 a12 a13

0a11a22 − a21a12

a11

a11a23 − a21a13

a11

0a11a32 − a31a12

a11

a11a33 − a31a13

a11

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

86 Chapter 2 Determinants

The matrix on the right will be row equivalent to I if and only if

a11

∣∣∣∣∣∣∣∣a11a22 − a21a12

a11

a11a23 − a21a13

a11

a11a32 − a31a12

a11

a11a33 − a31a13

a11

∣∣∣∣∣∣∣∣ �= 0

Although the algebra is somewhat messy, this condition can be simplified to

a11a22a33 − a11a32a23 − a12a21a33 + a12a31a23

+ a13a21a32 − a13a31a22 �= 0 (2)

Thus, if we define

det(A) = a11a22a33 − a11a32a23 − a12a21a33 (3)

+ a12a31a23 + a13a21a32 − a13a31a22

then, for the case a11 �= 0, the matrix will be nonsingular if and only if det(A) �= 0.What if a11 = 0? Consider the following possibilities:

(i) a11 = 0, a21 �= 0

(ii) a11 = a21 = 0, a31 �= 0

(iii) a11 = a21 = a31 = 0

In case (i), it is not difficult to show that A is row equivalent to I if and only if

−a12a21a33 + a12a31a23 + a13a21a32 − a13a31a22 �= 0

But this condition is the same as condition (2) with a11 = 0. The details of case (i) areleft as an exercise for the reader (see Exercise 7 at the end of this section).

In case (ii), it follows that

A =⎧⎪⎪⎪⎪⎪⎩

0 a12 a13

0 a22 a23

a31 a32 a33

⎫⎪⎪⎪⎪⎪⎭is row equivalent to I if and only if

a31(a12a23 − a22a13) �= 0

Again, this is a special case of condition (2) with a11 = a21 = 0.Clearly, in case (iii) the matrix A cannot be row equivalent to I and hence must

be singular. In this case, if we set a11, a21, and a31 equal to 0 in formula (3), the resultwill be det(A) = 0.

In general, then, formula (2) gives a necessary and sufficient condition for a 3 × 3matrix A to be nonsingular (regardless of the value of a11).

We would now like to define the determinant of an n × n matrix. To see how to dothis, note that the determinant of a 2 × 2 matrix

A =⎧⎪⎩a11 a12

a21 a22

⎫⎪⎭


can be defined in terms of the two 1 × 1 matrices

M11 = (a22) and M12 = (a21)

The matrix M11 is formed from A by deleting its first row and first column, and M12 isformed from A by deleting its first row and second column.

The determinant of A can be expressed in the form

det(A) = a11a22 − a12a21 = a11 det(M11) − a12 det(M12) (4)

For a 3 × 3 matrix A, we can rewrite equation (3) in the form

det(A) = a11(a22a33 − a32a23) − a12(a21a33 − a31a23) + a13(a21a32 − a31a22)

For j = 1, 2, 3, let M1 j denote the 2×2 matrix formed from A by deleting its first rowand j th column. The determinant of A can then be represented in the form

det(A) = a11 det(M11) − a12 det(M12) + a13 det(M13) (5)

where

M11 =⎧⎪⎩a22 a23

a32 a33

⎫⎪⎭ , M12 =⎧⎪⎩a21 a23

a31 a33

⎫⎪⎭ , M13 =⎧⎪⎩a21 a22

a31 a32

⎫⎪⎭To see how to generalize (4) and (5) to the case n > 3, we introduce the following

definition:

Definition Let A = (ai j ) be an n × n matrix, and let Mi j denote the (n − 1) × (n − 1) matrixobtained from A by deleting the row and column containing ai j . The determinantof Mi j is called the minor of ai j . We define the cofactor Ai j of ai j by

Ai j = (−1)i+ j det(Mi j )

In view of this definition, for a 2 × 2 matrix A, we may rewrite equation (4) in theform

det(A) = a11 A11 + a12 A12 (n = 2) (6)

Equation (6) is called the cofactor expansion of det(A) along the first row of A. Notethat we could also write

det(A) = a21(−a12) + a22a11 = a21 A21 + a22 A22 (7)

Equation (7) expresses det(A) in terms of the entries of the second row of A and theircofactors. Actually, there is no reason that we must expand along a row of the matrix;the determinant could just as well be represented by the cofactor expansion along oneof the columns:

det(A) = a11a22 + a21(−a12)

= a11 A11 + a21 A21 (first column)

det(A) = a12(−a21) + a22a11

= a12 A12 + a22 A22 (second column)


For a 3 × 3 matrix A, we have

det(A) = a11 A11 + a12 A12 + a13 A13 (8)

Thus, the determinant of a 3 × 3 matrix can be defined in terms of the elements in thefirst row of the matrix and their corresponding cofactors.

EXAMPLE 1 If

A =⎧⎪⎪⎪⎪⎪⎩

2 5 43 1 25 4 6

⎫⎪⎪⎪⎪⎪⎭then

det(A) = a11 A11 + a12 A12 + a13 A13

= (−1)2a11 det(M11) + (−1)3a12 det(M12) + (−1)4a13 det(M13)

= 2

∣∣∣∣1 24 6

∣∣∣∣ − 5

∣∣∣∣3 25 6

∣∣∣∣ + 4

∣∣∣∣3 15 4

∣∣∣∣= 2(6 − 8) − 5(18 − 10) + 4(12 − 5)

= −16

As in the case of 2×2 matrices, the determinant of a 3×3 matrix can be representedas a cofactor expansion using any row or column. For example, equation (3) can berewritten in the form

det(A) = a12a31a23 − a13a31a22 − a11a32a23 + a13a21a32 + a11a22a33 − a12a21a33

= a31(a12a23 − a13a22) − a32(a11a23 − a13a21) + a33(a11a22 − a12a21)

= a31 A31 + a32 A32 + a33 A33

This is the cofactor expansion along the third row of A.

EXAMPLE 2 Let A be the matrix in Example 1. The cofactor expansion of det(A) along the secondcolumn is given by

det(A) = −5

∣∣∣∣3 25 6

∣∣∣∣ + 1

∣∣∣∣2 45 6

∣∣∣∣ − 4

∣∣∣∣2 43 2

∣∣∣∣= −5(18 − 10) + 1(12 − 20) − 4(4 − 12) = −16

The determinant of a 4 × 4 matrix can be defined in terms of a cofactor expansionalong any row or column. To compute the value of the 4 × 4 determinant, we wouldhave to evaluate four 3 × 3 determinants.


Definition The determinant of an n × n matrix A, denoted det(A), is a scalar associated withthe matrix A that is defined inductively as

det(A) ={

a11 if n = 1a11 A11 + a12 A12 + · · · + a1n A1n if n > 1

whereA1 j = (−1)1+ j det(M1 j ) j = 1, . . . , n

are the cofactors associated with the entries in the first row of A.

As we have seen, it is not necessary to limit ourselves to using the first row for thecofactor expansion. We state the following theorem without proof:

Theorem 2.1.1 If A is an n×n matrix with n ≥ 2, then det(A) can be expressed as a cofactor expansionusing any row or column of A; that is,

det(A) = ai1 Ai1 + ai2 Ai2 + · · · + ain Ain

= a1 j A1 j + a2 j A2 j + · · · + anj Anj

for i = 1, . . . , n and j = 1, . . . , n.

The cofactor expansion of a 4×4 determinant will involve four 3×3 determinants.We can often save work by expanding along the row or column that contains the mostzeros. For example, to evaluate ∣∣∣∣∣∣∣

0 2 3 00 4 5 00 1 0 32 0 1 3

∣∣∣∣∣∣∣we would expand down the first column. The first three terms will drop out, leaving

−2

∣∣∣∣∣∣2 3 04 5 01 0 3

∣∣∣∣∣∣ = −2 · 3 ·∣∣∣∣2 34 5

∣∣∣∣ = 12

For n ≤ 3, we have seen that an n × n matrix A is nonsingular if and only ifdet(A) �= 0. In the next section, we will show that this result holds for all valuesof n. In that section we also look at the effect of row operations on the value of thedeterminant, and we will make use of row operations to derive a more efficient methodfor computing the value of a determinant.

We close this section with three theorems that are simple consequences of thecofactor expansion definition. The proofs of the last two theorems are left for thereader (see Exercises 8, 9, and 10 at the end of this section).

Theorem 2.1.2 If A is an n × n matrix, then det(AT ) = det(A).


Proof The proof is by induction on n. Clearly, the result holds if n = 1, since a 1 × 1 matrixis necessarily symmetric. Assume that the result holds for all k × k matrices and thatA is a (k + 1) × (k + 1) matrix. Expanding det(A) along the first row of A, we get

det(A) = a11 det(M11) − a12 det(M12) + − · · · ± a1,k+1 det(M1,k+1)

Since the Mi j ’s are all k × k matrices, it follows from the induction hypothesis that

det(A) = a11 det(MT11) − a12 det(MT

12) + − · · · ± a1,k+1 det(MT1,k+1) (9)

The right-hand side of (9) is just the expansion by minors of det(AT ) using the firstcolumn of AT . Therefore,

det(AT ) = det(A)

Theorem 2.1.3 If A is an n × n triangular matrix, then the determinant of A equals the product of thediagonal elements of A.

Proof In view of Theorem 2.1.2, it suffices to prove the theorem for lower triangular matrices.The result follows easily using the cofactor expansion and induction on n. The detailsare left for the reader (see Exercise 8 at the end of the section).

Theorem 2.1.4 Let A be an n × n matrix.

(i) If A has a row or column consisting entirely of zeros, then det(A) = 0.

(ii) If A has two identical rows or two identical columns, then det(A) = 0.

Both of these results can be easily proved with the use of the cofactor expansion.The proofs are left for the reader (see Exercises 9 and 10 at the end of the section).

In the next section, we look at the effect of row operations on the value of thedeterminant. This will allow us to make use of Theorem 2.1.3 to derive a more efficientmethod for computing the value of a determinant.

SECTION 2.1 EXERCISES1. Let

A =⎧⎪⎪⎪⎪⎪⎩

3 2 41 −2 32 3 2

⎫⎪⎪⎪⎪⎪⎭(a) Find the values of det(M21), det(M22), and

det(M23).

(b) Find the values of A21, A22, and A23.

(c) Use your answers from part (b) to computedet(A).

2. Use determinants to determine whether the follow-ing 2 × 2 matrices are nonsingular:

(a)⎧⎪⎩3 5

2 4

⎫⎪⎭ (b)⎧⎪⎩3 6

2 4

⎫⎪⎭(c)

⎧⎪⎩3 −62 4

⎫⎪⎭3. Evaluate the following determinants:

(a)

∣∣∣∣ 3 5−2 −3

∣∣∣∣ (b)

∣∣∣∣ 5 −2−8 4

∣∣∣∣(c)

∣∣∣∣∣∣3 1 22 4 52 4 5

∣∣∣∣∣∣ (d)

∣∣∣∣∣∣4 3 03 1 25 −1 −4

∣∣∣∣∣∣

2.2 Properties of Determinants 91

(e)

∣∣∣∣∣∣1 3 24 1 −22 1 3

∣∣∣∣∣∣ (f)

∣∣∣∣∣∣2 −1 21 3 25 1 6

∣∣∣∣∣∣(g)

∣∣∣∣∣∣∣2 0 0 10 1 0 01 6 2 01 1 −2 3

∣∣∣∣∣∣∣

(h)

∣∣∣∣∣∣∣2 1 2 13 0 1 1

−1 2 −2 1−3 2 3 1

∣∣∣∣∣∣∣4. Evaluate the following determinants by inspection:

(a)

∣∣∣∣3 52 4

∣∣∣∣ (b)

∣∣∣∣∣∣2 0 04 1 07 3 −2

∣∣∣∣∣∣(c)

∣∣∣∣∣∣3 0 02 1 11 2 2

∣∣∣∣∣∣ (d)

∣∣∣∣∣∣∣4 0 2 15 0 4 22 0 3 41 0 2 3

∣∣∣∣∣∣∣5. Evaluate the following determinant. Write your an-

swer as a polynomial in x .

∣∣∣∣∣∣a − x b c

1 −x 00 1 −x

∣∣∣∣∣∣6. Find all values of λ for which the following deter-

minant will equal 0:

∣∣∣∣2 − λ 43 3 − λ

∣∣∣∣

7. Let A be a 3 × 3 matrix with a11 = 0 and a21 �= 0.Show that A is row equivalent to I if and only if

− a12a21a33 + a12a31a23

+ a13a21a32 − a13a31a22 �= 0

8. Write out the details of the proof of Theorem 2.1.3.

9. Prove that if a row or a column of an n × n matrixA consists entirely of zeros, then det(A) = 0.

10. Use mathematical induction to prove that if A is an(n + 1) × (n + 1) matrix with two identical rows,then det(A) = 0.

11. Let A and B be 2 × 2 matrices.(a) Does det(A + B) = det(A) + det(B)?(b) Does det(AB) = det(A) det(B)?(c) Does det(AB) = det(BA)?Justify your answers.

12. Let A and B be 2 × 2 matrices and let

C =⎧⎪⎩a11 a12

b21 b22

⎫⎪⎭ , D =⎧⎪⎩b11 b12

a21 a22

⎫⎪⎭ ,

E =⎧⎪⎩ 0 α

β 0

⎫⎪⎭(a) Show that det(A + B) = det(A) + det(B) +

det(C) + det(D).(b) Show that if B = EA then det(A + B) =

det(A) + det(B).

13. Let A be a symmetric tridiagonal matrix (i.e., A issymmetric and ai j = 0 whenever |i − j | > 1). LetB be the matrix formed from A by deleting the firsttwo rows and columns. Show that

det(A) = a11 det(M11) − a212 det(B)

2.2 Properties of Determinants

In this section, we consider the effects of row operations on the determinant of a matrix.Once these effects have been established, we will prove that a matrix A is singularif and only if its determinant is zero, and we will develop a method for evaluatingdeterminants by using row operations. Also, we will establish an important theoremabout the determinant of the product of two matrices. We begin with the followinglemma:

Lemma 2.2.1 Let A be an n × n matrix. If A jk denotes the cofactor of a jk for k = 1, . . . , n, then

ai1 A j1 + ai2 A j2 + · · · + ain A jn ={

det(A) if i = j0 if i �= j

(1)


Proof If i = j , (1) is just the cofactor expansion of det(A) along the i th row of A. To prove(1) in the case i �= j , let A∗ be the matrix obtained by replacing the j th row of A bythe i th row of A:

A∗ =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

a11 a12 · · · a1n...

ai1 ai2 · · · ain...

ai1 ai2 · · · ain...

an1 an2 · · · ann

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

j th row

Since two rows of A∗ are the same, its determinant must be zero. It follows from thecofactor expansion of det(A∗) along the j th row that

0 = det(A∗) = ai1 A∗j1 + ai2 A∗

j2 + · · · + ain A∗jn

= ai1 A j1 + ai2 A j2 + · · · + ain A jn

Let us now consider the effects of each of the three row operations on the value ofthe determinant.

Row Operation I

Two rows of A are interchanged.

If A is a 2 × 2 matrix and

E =⎧⎪⎩0 1

1 0

⎫⎪⎭then

det(E A) =∣∣∣∣a21 a22

a11 a12

∣∣∣∣ = a21a12 − a22a11 = − det(A)

For n > 2, let Ei j be the elementary matrix that switches rows i and j of A. It isa simple induction proof to show that det(Ei j A) = − det(A). We illustrate the ideabehind the proof for the case n = 3. Suppose that the first and third rows of a 3 × 3matrix A have been interchanged. Expanding det(E13 A) along the second row andmaking use of the result for 2 × 2 matrices, we see that

det(E13 A) =∣∣∣∣∣∣a31 a32 a33

a21 a22 a23

a11 a12 a13

∣∣∣∣∣∣= −a21

∣∣∣∣a32 a33

a12 a13

∣∣∣∣ + a22

∣∣∣∣a31 a33

a11 a13

∣∣∣∣ − a23

∣∣∣∣a31 a32

a11 a12

∣∣∣∣= a21

∣∣∣∣a12 a13

a32 a33

∣∣∣∣ − a22

∣∣∣∣a11 a13

a31 a33

∣∣∣∣ + a23

∣∣∣∣a11 a12

a31 a32

∣∣∣∣= − det(A)


In general, if A is an n × n matrix and Ei j is the n × n elementary matrix formed byinterchanging the i th and j th rows of I , then

det(Ei j A) = − det(A)

In particular,det(Ei j ) = det(Ei j I ) = − det(I ) = −1

Thus, for any elementary matrix E of type I,

det(EA) = − det(A) = det(E) det(A)

Row Operation II

A row of A is multiplied by a nonzero constant.

Let E denote the elementary matrix of type II formed from I by multiplying thei th row by the nonzero constant α. If det(EA) is expanded by cofactors along the i throw, then

det(E A) = αai1 Ai1 + αai2 Ai2 + · · · + αain Ain

= α(ai1 Ai1 + ai2 Ai2 + · · · + ain Ain)

= α det(A)

In particular,det(E) = det(E I ) = α det(I ) = α

and hence,det(EA) = α det(A) = det(E) det(A)

Row Operation III

A multiple of one row is added to another row.

Let E be the elementary matrix of type III formed from I by adding c times the i throw to the j th row. Since E is triangular and its diagonal elements are all 1, it followsthat det(E) = 1. We will show that

det(EA) = det(A) = det(E) det(A)

If det(EA) is expanded by cofactors along the j th row, it follows from Lemma 2.2.1that

det(E A) = (a j1 + cai1)A j1 + (a j2 + cai2)A j2 + · · · + (a jn +cain)A jn

= (a j1 A j1 + · · · + a jn A jn) + c(ai1 A j1 + · · · + ain A jn)

= det(A)

Thus,det(E A) = det(A) = det(E) det(A)


SUMMARY In summation, if E is an elementary matrix, then

det(E A) = det(E) det(A)

where

det(E) =

⎧⎪⎨⎪⎩

−1 if E is of type Iα �= 0 if E is of type II

1 if E is of type III

(2)

Similar results hold for column operations. Indeed, if E is an elementary matrix,then ET is also an elementary matrix (see Exercise 8 at the end of this section) and

det(AE) = det((AE)T

) = det(ET AT

)= det

(ET

)det

(AT

) = det(E) det(A)

Thus, the effects that row or column operations have on the value of the determinantcan be summarized as follows:

I. Interchanging two rows (or columns) of a matrix changes the sign of thedeterminant.

II. Multiplying a single row or column of a matrix by a scalar has the effectof multiplying the value of the determinant by that scalar.

III. Adding a multiple of one row (or column) to another does not change thevalue of the determinant.

Note

As a consequence of III, if one row (or column) of a matrix is a multiple of another,the determinant of the matrix must equal zero.

Main Results

We can now make use of the effects of row operations on determinants to prove two ma-jor theorems and to establish a simpler method of computing determinants. It followsfrom (2) that all elementary matrices have nonzero determinants. This observation canbe used to prove the following theorem:

Theorem 2.2.2 An n × n matrix A is singular if and only if

det(A) = 0

Proof The matrix A can be reduced to row echelon form with a finite number of row opera-tions. Thus,

U = Ek Ek−1 · · · E1 A

where U is in row echelon form and the Ei ’s are all elementary matrices. It follows


that

det(U ) = det(Ek Ek−1 · · · E1 A)

= det(Ek) det(Ek−1) · · · det(E1) det(A)

Since the determinants of the Ei ’s are all nonzero, it follows that det(A) = 0 if andonly if det(U ) = 0. If A is singular, then U has a row consisting entirely of zeros,and hence det(U ) = 0. If A is nonsingular, then U is triangular with 1’s along thediagonal, and hence det(U ) = 1.

From the proof of Theorem 2.2.2, we can obtain a method for computing det(A).We reduce A to row echelon form:

U = Ek Ek−1 · · · E1 A

If the last row of U consists entirely of zeros, A is singular and det(A) = 0. Otherwise,A is nonsingular and

det(A) = [det(Ek) det(Ek−1) · · · det(E1)

]−1

Actually, if A is nonsingular, it is simpler to reduce A to triangular form. This can bedone using only row operations I and III. Thus,

T = Em Em−1 · · · E1 A

and hence,det(A) = ± det(T ) = ±t11t22 · · · tnn

where the tii ’s are the diagonal entries of T . The sign will be positive if row operationI has been used an even number of times and negative otherwise.

EXAMPLE 1 Evaluate ∣∣∣∣∣∣2 1 34 2 16 −3 4

∣∣∣∣∣∣Solution

∣∣∣∣∣∣2 1 34 2 16 −3 4

∣∣∣∣∣∣ =∣∣∣∣∣∣2 1 30 0 −50 −6 −5

∣∣∣∣∣∣ = (−1)

∣∣∣∣∣∣2 1 30 −6 −50 0 −5

∣∣∣∣∣∣= (−1)(2)(−6)(−5)

= −60

We now have two methods for evaluating the determinant of an n × n matrix A.If n > 3 and A has nonzero entries, elimination is the most efficient method, in thesense that it involves fewer arithmetic operations. In Table 1, the number of arithmeticoperations involved in each method is given for n = 2, 3, 4, 5, 10. It is not difficult


Table 1

Cofactors Elimination

Multiplicationsn Additions Multiplications Additions and Divisions

2

3

4

5

10

1

5

23

119

3,628,799

2

9

40

205

6,235,300

1

5

14

30

285

3

10

23

44

339

to derive general formulas for the number of operations in each of the methods (seeExercises 20 and 21 at the end of the section).

We have seen that, for any elementary matrix E ,

det(EA) = det(E) det(A) = det(AE)

This is a special case of the following theorem:

Theorem 2.2.3 If A and B are n × n matrices, then

det(AB) = det(A) det(B)

Proof If B is singular, it follows from Theorem 1.5.2 that AB is also singular (see Exercise 14of Chapter 1, Section 5), and therefore,

det(AB) = 0 = det(A) det(B)

If B is nonsingular, B can be written as a product of elementary matrices. We havealready seen that the result holds for elementary matrices. Thus,

det(AB) = det(AEk Ek−1 · · · E1)

= det(A) det(Ek) det(Ek−1) · · · det(E1)

= det(A) det(Ek Ek−1 · · · E1)

= det(A) det(B)

If A is singular, the computed value of det(A) using exact arithmetic must be0. However, this result is unlikely if the computations are done by computer. Sincecomputers use a finite number system, roundoff errors are usually unavoidable. Con-sequently, it is more likely that the computed value of det(A) will only be near 0.Because of roundoff errors, it is virtually impossible to determine computationallywhether a matrix is exactly singular. In computer applications, it is often more mean-ingful to ask whether a matrix is “close” to being singular. In general, the value ofdet(A) is not a good indicator of nearness to singularity. In Section 5 of Chapter 6, wewill discuss how to determine whether a matrix is close to being singular.


SECTION 2.2 EXERCISES1. Evaluate each of the following determinants by in-

spection:

(a)

∣∣∣∣∣∣0 0 30 4 12 3 1

∣∣∣∣∣∣(b)

∣∣∣∣∣∣∣1 1 1 30 3 1 10 0 2 2

−1 −1 −1 2

∣∣∣∣∣∣∣(c)

∣∣∣∣∣∣∣0 0 0 11 0 0 00 1 0 00 0 1 0

∣∣∣∣∣∣∣2. Let

A =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎩0 1 2 31 1 1 1

−2 −2 3 31 2 −2 −3

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎭(a) Use the elimination method to evaluate det(A).

(b) Use the value of det(A) to evaluate

∣∣∣∣∣∣∣0 1 2 3

−2 −2 3 31 2 −2 −31 1 1 1

∣∣∣∣∣∣∣+

∣∣∣∣∣∣∣0 1 2 31 1 1 1

−1 −1 4 42 3 −1 −2

∣∣∣∣∣∣∣3. For each of the following, compute the determinant

and state whether the matrix is singular or nonsin-gular:

(a)⎧⎪⎩3 1

6 2

⎫⎪⎭ (b)⎧⎪⎩3 1

4 2

⎫⎪⎭(c)

⎧⎪⎪⎪⎪⎪⎩3 3 10 1 20 2 3

⎫⎪⎪⎪⎪⎪⎭ (d)

⎧⎪⎪⎪⎪⎪⎩2 1 14 3 52 1 2

⎫⎪⎪⎪⎪⎪⎭(e)

⎧⎪⎪⎪⎪⎪⎩2 −1 3

−1 2 −21 4 0

⎫⎪⎪⎪⎪⎪⎭

(f)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎩1 1 1 12 −1 3 20 1 2 10 0 7 3

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎭

4. Find all possible choices of c that would make thefollowing matrix singular:⎧⎪⎪⎪⎪⎪⎩

1 1 11 9 c1 c 3

⎫⎪⎪⎪⎪⎪⎭5. Let A be an n × n matrix and α a scalar. Show that

det(αA) = αn det(A)

6. Let A be a nonsingular matrix. Show that

det(A−1) = 1

det(A)

7. Let A and B be 3×3 matrices with det(A) = 4 anddet(B) = 5. Find the value of(a) det(AB) (b) det(3A)

(c) det(2AB) (d) det(A−1 B)

8. Show that if E is an elementary matrix, then E T isan elementary matrix of the same type as E .

9. Let E1, E2, and E3 be 3 × 3 elementary matrices oftypes I, II, and III, respectively, and let A be a 3×3matrix with det(A) = 6. Assume, additionally, thatE2 was formed from I by multiplying its secondrow by 3. Find the values of each of the following:(a) det(E1 A) (b) det(E2 A)

(c) det(E3 A) (d) det(AE1)

(e) det(E21) (f) det(E1 E2 E3)

10. Let A and B be row equivalent matrices, and sup-pose that B can be obtained from A by using onlyrow operations I and III. How do the values ofdet(A) and det(B) compare? How will the valuescompare if B can be obtained from A by using onlyrow operation III? Explain your answers.

11. Let A be an n × n matrix. Is it possible for A2+I =O in the case where n is odd? Answer the samequestion in the case where n is even.

12. Consider the 3 × 3 Vandermonde matrix

V =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎩1 x1 x2

1

1 x2 x22

1 x3 x23

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎭(a) Show that det(V ) = (x2−x1)(x3−x1)(x3−x2).

[Hint: Make use of row operation III.]

(b) What conditions must the scalars x1, x2, and x3

satisfy in order for V to be nonsingular?


13. Suppose that a 3×3 matrix A factors into a product⎧⎪⎪⎪⎪⎪⎩1 0 0

l21 1 0l31 l32 1

⎫⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎩

u11 u12 u13

0 u22 u23

0 0 u33

⎫⎪⎪⎪⎪⎪⎭Determine the value of det(A).

14. Let A and B be n ×n matrices. Prove that the prod-uct AB is nonsingular if and only if A and B areboth nonsingular.

15. Let A and B be n × n matrices. Prove that ifAB = I , then BA = I . What is the significanceof this result in terms of the definition of a nonsin-gular matrix?

16. A matrix A is said to be skew symmetric ifAT = −A. For example,

A =⎧⎪⎩ 0 1

−1 0

⎫⎪⎭is skew symmetric, since

AT =⎧⎪⎩0 −1

1 0

⎫⎪⎭ = −A

If A is an n × n skew-symmetric matrix and n isodd, show that A must be singular.

17. Let A be a nonsingular n ×n matrix with a nonzerocofactor Ann , and set

c = det(A)

Ann

Show that if we subtract c from ann , then the result-ing matrix will be singular.

18. Let A be a k × k matrix and let B be an(n − k) × (n − k) matrix. Let

E =⎧⎪⎩ Ik O

O B

⎫⎪⎭ , F =⎧⎪⎩ A O

O In−k

⎫⎪⎭ ,

C =⎧⎪⎩ A O

O B

⎫⎪⎭where Ik and In−k are the k × k and(n − k) × (n − k) identity matrices.(a) Show that det(E) = det(B).(b) Show that det(F) = det(A).(c) Show that det(C) = det(A) det(B).

19. Let A and B be k × k matrices and let

M =⎧⎪⎩ O B

A O

⎫⎪⎭Show that det(M) = (−1)k det(A) det(B).

20. Show that evaluating the determinant of an n × nmatrix by cofactors involves (n! − 1) additions andn−1∑k=1

n!/k! multiplications.

21. Show that the elimination method of computingthe value of the determinant of an n × n ma-trix involves [n(n − 1)(2n − 1)]/6 additions and[(n − 1)(n2 + n + 3)]/3 multiplications and di-visions. [Hint: At the ith step of the reductionprocess, it takes n − i divisions to calculate themultiples of the ith row that are to be subtractedfrom the remaining rows below the pivot. We mustthen calculate new values for the (n − i)2 entries inrows i +1 through n and columns i +1 through n.]

2.3 Additional Topics and Applications

In this section, we learn a method for computing the inverse of a nonsingular matrixA using determinants and we learn a method for solving linear systems using determi-nants. Both methods depend on Lemma 2.2.1. We also show how to use determinantsto define the cross product of two vectors. The cross product is useful in physics appli-cations involving the motion of a particle in 3-space.

2.3 Additional Topics and Applications 99

The Adjoint of a Matrix

Let A be an n × n matrix. We define a new matrix called the adjoint of A by

adj A =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩A11 A21 · · · An1A12 A22 · · · An2

...A1n A2n · · · Ann

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭Thus, to form the adjoint, we must replace each term by its cofactor and then transposethe resulting matrix. By Lemma 2.2.1,

ai1 A j1 + ai2 A j2 + · · · + ain A jn ={

det(A) if i = j0 if i �= j

and it follows that

A(adj A) = det(A)I

If A is nonsingular, det(A) is a nonzero scalar, and we may write

A

(1

det(A)adj A

)= I

Thus,

A−1 = 1

det(A)adj A when det(A) �= 0

EXAMPLE 1 For a 2 × 2 matrix,

adj A =⎧⎪⎩ a22 −a12

−a21 a11

⎫⎪⎭If A is nonsingular, then

A−1 = 1

a11a22 − a12a21

⎧⎪⎩ a22 −a12

−a21 a11

⎫⎪⎭EXAMPLE 2 Let

A =⎧⎪⎪⎪⎪⎪⎩

2 1 23 2 21 2 3

⎫⎪⎪⎪⎪⎪⎭Compute adj A and A−1.


Solution

adj A =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

∣∣∣∣2 22 3

∣∣∣∣ −∣∣∣∣3 21 3

∣∣∣∣∣∣∣∣3 21 2

∣∣∣∣−

∣∣∣∣1 22 3

∣∣∣∣∣∣∣∣2 21 3

∣∣∣∣ −∣∣∣∣2 11 2

∣∣∣∣∣∣∣∣1 22 2

∣∣∣∣ −∣∣∣∣2 23 2

∣∣∣∣∣∣∣∣2 13 2

∣∣∣∣

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

T

=⎧⎪⎪⎪⎪⎪⎩

2 1 −2−7 4 2

4 −3 1

⎫⎪⎪⎪⎪⎪⎭

A−1 = 1

det(A)adj A = 1

5

⎧⎪⎪⎪⎪⎪⎩2 1 −2

−7 4 24 −3 1

⎫⎪⎪⎪⎪⎪⎭Using the formula

A−1 = 1

det(A)adj A

we can derive a rule for representing the solution to the system Ax = b in terms ofdeterminants.

Cramer’s Rule

Theorem 2.3.1 Cramer's RuleLet A be an n × n nonsingular matrix, and let b ∈ R

n . Let Ai be the matrix obtainedby replacing the i th column of A by b. If x is the unique solution of Ax = b, then

xi = det(Ai )

det(A)for i = 1, 2, . . . , n

Proof Since

x = A−1b = 1

det(A)(adj A)b

it follows that

xi = b1 A1i + b2 A2i + · · · + bn Ani

det(A)

= det(Ai )

det(A)

EXAMPLE 3 Use Cramer’s rule to solve

x1 + 2x2 + x3 = 52x1 + 2x2 + x3 = 6

x1 + 2x2 + 3x3 = 9


Solution

det(A) =∣∣∣∣∣∣1 2 12 2 11 2 3

∣∣∣∣∣∣ = −4 det(A1) =∣∣∣∣∣∣5 2 16 2 19 2 3

∣∣∣∣∣∣ = −4

det(A2) =∣∣∣∣∣∣1 5 12 6 11 9 3

∣∣∣∣∣∣ = −4 det(A3) =∣∣∣∣∣∣1 2 52 2 61 2 9

∣∣∣∣∣∣ = −8

Therefore,

x1 = −4

−4= 1, x2 = −4

−4= 1, x3 = −8

−4= 2

Cramer’s rule gives us a convenient method for writing the solution of an n × nsystem of linear equations in terms of determinants. To compute the solution, however,we must evaluate n + 1 determinants of order n. Evaluating even two of these deter-minants generally involves more computation than solving the system using Gaussianelimination.

APPLICATION 1 Coded Messages

A common way of sending a coded message is to assign an integer value to each letterof the alphabet and to send the message as a string of integers. For example, themessage

SEND MONEY

might be coded as5, 8, 10, 21, 7, 2, 10, 8, 3

Here the S is represented by a 5, the E by an 8, and so on. Unfortunately, this type ofcode is generally easy to break. In a longer message we might be able to guess whichletter is represented by a number on the basis of the relative frequency of occurrenceof that number. For example, if 8 is the most frequently occurring number in thecoded message, then it is likely that it represents the letter E, the letter that occurs mostfrequently in the English language.

We can disguise the message further by using matrix multiplications. If A is amatrix whose entries are all integers and whose determinant is ±1, then, since A−1 =± adj A, the entries of A−1 will be integers. We can use such a matrix to transform themessage. The transformed message will be more difficult to decipher. To illustrate thetechnique, let

A =⎧⎪⎪⎪⎪⎪⎩

1 2 12 5 32 3 2

⎫⎪⎪⎪⎪⎪⎭The coded message is put into the columns of a matrix B having three rows:

B =⎧⎪⎪⎪⎪⎪⎩

5 21 108 7 8

10 2 3

⎫⎪⎪⎪⎪⎪⎭


The product

AB =⎧⎪⎪⎪⎪⎪⎩

1 2 12 5 32 3 2

⎫⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎩

5 21 108 7 8

10 2 3

⎫⎪⎪⎪⎪⎪⎭ =⎧⎪⎪⎪⎪⎪⎩

31 37 2980 83 6954 67 50

⎫⎪⎪⎪⎪⎪⎭gives the coded message to be sent:

31, 80, 54, 37, 83, 67, 29, 69, 50

The person receiving the message can decode it by multiplying by A−1:⎧⎪⎪⎪⎪⎪⎩1 −1 12 0 −1

−4 1 1

⎫⎪⎪⎪⎪⎪⎭⎧⎪⎪⎪⎪⎪⎩

31 37 2980 83 6954 67 50

⎫⎪⎪⎪⎪⎪⎭ =⎧⎪⎪⎪⎪⎪⎩

5 21 108 7 8

10 2 3

⎫⎪⎪⎪⎪⎪⎭To construct a coding matrix A, we can begin with the identity I and successively

apply row operation III, being careful to add integer multiples of one row to another.Row operation I can also be used. The resulting matrix A will have integer entries, andsince

det(A) = ± det(I ) = ±1

A−1 will also have integer entries.

References1. Hansen, Robert, Two-Year College Mathematics Journal, 13(1), 1982.

The Cross Product

Given two vectors x and y in R3, one can define a third vector, the cross product,

denoted x × y, by

x × y =⎧⎪⎪⎪⎪⎪⎩

x2 y3 − y2x3

y1x3 − x1 y3

x1 y2 − y1x2

⎫⎪⎪⎪⎪⎪⎭ (1)

If C is any matrix of the form

C =⎧⎪⎪⎪⎪⎪⎩

w1 w2 w3

x1 x2 x3

y1 y2 y3

⎫⎪⎪⎪⎪⎪⎭then

x × y = C11e1 + C12e2 + C13e3 =⎧⎪⎪⎪⎪⎪⎩

C11

C12

C13

⎫⎪⎪⎪⎪⎪⎭Expanding det(C) by cofactors along the first row, we see that

det(C) = w1C11 + w2C12 + w3C13 = wT (x × y)


In particular, if we choose w = x or w = y, then the matrix C will have two identicalrows, and hence its determinant will be 0. We then have

xT (x × y) = yT (x × y) = 0 (2)

In calculus books, it is standard to use row vectors

x = (x1, x2, x3) and y = (y1, y2, y3)

and to define the cross product to be the row vector

x × y = (x2 y3 − y2x3)i − (x1 y3 − y1x3)j + (x1 y2 − y1x2)k

where i, j, and k are the row vectors of the 3 × 3 identity matrix. If one uses i, j, and kin place of w1, w2, and w3, respectively, in the first row of the matrix M , then the crossproduct can be written as a determinant:

x × y =∣∣∣∣∣∣

i j kx1 x2 x3

y1 y2 y3

∣∣∣∣∣∣In linear algebra courses, it is generally more standard to view x, y, and x×y as columnvectors. In this case, we can represent the cross product in terms of the determinantof a matrix whose entries in the first row are e1, e2, and e3, the column vectors of the3 × 3 identity matrix:

x × y =∣∣∣∣∣∣e1 e2 e3

x1 x2 x3

y1 y2 y3

∣∣∣∣∣∣The relation given in equation (2) has applications in Newtonian mechanics. In

particular, the cross product can be used to define a binormal direction, which Newtonused to derive the laws of motion for a particle in 3-space.

APPLICATION 2 Newtonian Mechanics

If x is a vector in either R2 or R

3, then we can define the length of x, denoted ‖x‖, by

‖x‖ = (xT x)12

A vector x is said to be a unit vector if ‖x‖ = 1. Unit vectors were used by Newtonto derive the laws of motion for a particle in either the plane or 3-space. If x and y arenonzero vectors in R

2, then the angle θ between the vectors is the smallest angle ofrotation necessary to rotate one of the two vectors clockwise so that it ends up in thesame direction as the other vector (see Figure 2.3.1).

A particle moving in a plane traces out a curve in the plane. The position of theparticle at any time t can be represented by a vector (x1(t), x2(t)). In describing themotion of a particle, Newton found it convenient to represent the position of vectors attime t as linear combinations of the vectors T(t) and N(t), where T(t) is a unit vectorin the direction of the tangent line to curve at the point (x1(t), x2(t)) and N(t) is a unitvector in the direction of a normal line (a line perpendicular to the tangent line) to thecurve at the given point (see Figure 2.3.2).


x

y

θ

Figure 2.3.1.

1.510.5

0.5

1

1.5

2

2.5

3

3.5

00 2 2.5 3 3.5

N(t)

T(t)

Figure 2.3.2.

In Chapter 5, we will show that if x and y are nonzero vectors and θ is the anglebetween the vectors, then

xT y = ‖x‖‖y‖ cos θ (3)

This equation can also be used to define the angle between nonzero vectors in R3.

It follows from (3) that the angle between the vectors is a right angle if and only ifxT y = 0. In this case, we say that the vectors x and y are orthogonal. In particular,since T(t) and N(t) are unit orthogonal vectors in R

2, we have ‖T(t)‖ = ‖N(t)‖ = 1and the angle between the vectors is π

2 . It follows from (3) that

T(t)T N(t) = 0

In Chapter 5, we will also show that if x and y are vectors in R3 and θ is the angle

between the vectors, then‖x × y‖ = ‖x‖‖y‖ sin θ (4)

A particle moving in three dimensions will trace out a curve in 3-space. In thiscase, at time t the tangent and normal lines to the curve at the point (x1(t), x2(t))determine a plane in 3-space. However, in 3-space the motion is not restricted to aplane. To derive laws describing the motion, Newton needed to use a third vector, avector in a direction normal to the plane determined by T(t) and N(t). If z is anynonzero vector in the direction of the normal line to this plane, then the angle betweenthe vectors z and T(t) and the angle between z and N(t) should both be right angles.If we set

B(t) = T(t) × N(t) (5)

then it follows from (2) that B(t) is orthogonal to both T(t) and N(t) and hence is inthe direction of the normal line. Furthermore B(t) is a unit vector, since it follows


from (4) that

‖B(t)‖ = ‖T(t) × N(t)‖ = ‖T(t)‖‖N(t)‖ sinπ

2= 1

The vector B(t) defined by (5) is called the binormal vector (see Figure 2.3.3).

B(t)

N(t)

T(t)

Figure 2.3.3.

SECTION 2.3 EXERCISES1. For each of the following, compute (i) det(A),

(ii) adj A, and (iii) A−1:

(a) A =⎧⎪⎩1 2

3 −1

⎫⎪⎭ (b) A =⎧⎪⎩3 1

2 4

⎫⎪⎭(c) A =

⎧⎪⎪⎪⎪⎪⎩1 3 12 1 1

−2 2 −1

⎫⎪⎪⎪⎪⎪⎭(d) A =

⎧⎪⎪⎪⎪⎪⎩1 1 10 1 10 0 1

⎫⎪⎪⎪⎪⎪⎭2. Use Cramer’s rule to solve each of the following

systems:

(a) x1 + 2x2 = 3

3x1 − x2 = 1

(b) 2x1 + 3x2 = 2

3x1 + 2x2 = 5

(c) 2x1 + x2 − 3x3 = 0

4x1 + 5x2 + x3 = 8

−2x1 − x2 + 4x3 = 2

(d) x1 + 3x2 + x3 = 1

2x1 + x2 + x3 = 5

−2x1 + 2x2 − x3 = −8

(e) x1 + x2 = 0

x2 + x3 − 2x4 = 1

x1 + 2x3 + x4 = 0

x1 + x2 + x4 = 0

3. Given

A =⎧⎪⎪⎪⎪⎪⎩

1 2 10 4 31 2 2

⎫⎪⎪⎪⎪⎪⎭determine the (2, 3) entry of A−1 by computing aquotient of two determinants.

4. Let A be the matrix in Exercise 3. Compute thethird column of A−1 by using Cramer’s rule to solveAx = e3.

5. Let

A =⎧⎪⎪⎪⎪⎪⎩

1 2 32 3 43 4 5

⎫⎪⎪⎪⎪⎪⎭(a) Compute the determinant of A. Is A nonsingu-

lar?(b) Compute adj A and the product A adj A.

6. If A is singular, what can you say about the productA adj A?


7. Let Bj denote the matrix obtained by replacing thej th column of the identity matrix with a vectorb = (b1, . . . , bn)

T . Use Cramer’s rule to show that

b j = det(Bj ) for j = 1, . . . , n

8. Let A be a nonsingular n × n matrix with n > 1.Show that

det(adj A) = (det(A))n−1

9. Let A be a 4 × 4 matrix. If

adj A =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎩2 0 0 00 2 1 00 4 3 20 −2 −1 2

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎭(a) calculate the value of det(adj A). What should

the value of det(A) be? [Hint: Use the resultfrom Exercise 8.]

(b) find A.

10. Show that if A is nonsingular, then adj A is nonsin-gular and

(adj A)−1 = det(A−1)A = adj A−1

11. Show that if A is singular, then adj A is also singu-lar.

12. Show that if det(A) = 1, then

adj(adj A) = A

13. Suppose that Q is a matrix with the propertyQ−1 = QT . Show that

qi j = Qi j

det(Q)

14. In coding a message, a blank space was representedby 0, an A by 1, a B by 2, a C by 3, and so on. Themessage was transformed using the matrix

A =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎩−1 −1 2 0

1 1 −1 00 0 −1 11 0 0 −1

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎭and sent as

− 19, 19, 25, −21, 0, 18, −18, 15, 3, 10,

− 8, 3, −2, 20, −7, 12

What was the message?

15. Let x, y, and z be vectors in R3. Show each of the

following:(a) x × x = 0 (b) y × x = −(x × y)

(c) x × (y + z) = (x × y) + (x × z)

(d) zT (x × y) =∣∣∣∣∣∣

x1 x2 x3

y1 y2 y3

z1 z2 z3

∣∣∣∣∣∣16. Let x and y be vectors in R

3 and define the skew-symmetric matrix Ax by

Ax =⎧⎪⎪⎪⎪⎪⎩

0 −x3 x2

x3 0 −x1

−x2 x1 0

⎫⎪⎪⎪⎪⎪⎭(a) Show that x × y = Ax y.

(b) Show that y × x = ATx y.

Chapter Two Exercises

MATLAB EXERCISES

The first four exercises that follow involve integer matri-ces and illustrate some of the properties of determinantsthat were covered in this chapter. The last two exercisesillustrate some of the differences that may arise whenwe work with determinants in floating-point arithmetic.

In theory, the value of the determinant should tellus whether the matrix is nonsingular. However, if thematrix is singular and its determinant is computed usingfinite-precision arithmetic, then, because of roundoff er-rors, the computed value of the determinant may notequal zero. A computed value near zero does not nec-

essarily mean that the matrix is singular or even closeto being singular. Furthermore, a matrix may be nearlysingular and have a determinant that is not even closeto zero (see Exercise 6).

1. Generate random 5×5 matrices with integer entriesby setting

A = round(10 ∗ rand(5))

andB = round(20 ∗ rand(5)) − 10

Chapter Two Exercises 107

Use MATLAB to compute each of the pairs ofnumbers that follow. In each case, check whetherthe first number is equal to the second.(a) det(A) det(AT )

(b) det(A + B) det(A) + det(B)

(c) det(AB) det(A) det(B)

(d) det(AT BT ) det(AT ) det(BT )

(e) det(A−1) 1/ det(A)

(f) det(AB−1) det(A)/ det(B)

2. Are n × n magic squares nonsingular? Use theMATLAB command det(magic(n)) to computethe determinants of the magic squares matrices inthe cases n = 3, 4, . . . , 10. What seems to be hap-pening? Check the cases n = 24 and 25 to see ifthe pattern still holds.

3. Set A = round(10 ∗ rand(6)). In each of thefollowing, use MATLAB to compute a second ma-trix as indicated. State how the second matrix isrelated to A and compute the determinants of bothmatrices. How are the determinants related?(a) B = A; B(2, :) = A(1, :); B(1, :) = A(2, :)(b) C = A; C(3, :) = 4 ∗ A(3, :)(c) D = A; D(5, :) = A(5, :) + 2 ∗ A(4, :)

4. We can generate a random 6 × 6 matrix A whoseentries consist entirely of 0’s and 1’s by setting

A = round(rand(6))

(a) What percentage of these random 0–1 matricesare singular? You can estimate the percentagein MATLAB by setting

y = zeros(1, 100);and then generating 100 test matrices and set-ting y( j) = 1 if the j th matrix is singular and0 otherwise. The easy way to do this in MAT-LAB is to use a for loop. Generate the loop asfollows:

for j = 1 : 100

A = round(rand(6));y( j) = (det(A) == 0);

end

(Note: A semicolon at the end of a line sup-presses printout. It is recommended that youinclude one at the end of each line of calcula-tion that occurs inside a for loop.) To deter-mine how many singular matrices were gen-erated, use the MATLAB command sum(y).What percentage of the matrices generatedwere singular?

(b) For any positive integer n, we can generate arandom 6 × 6 matrix A whose entries are inte-gers from 0 to n by setting

A = round(n ∗ rand(6))

What percentage of random integer matricesgenerated in this manner will be singular ifn = 3? If n = 6? If n = 10? We can es-timate the answers to these questions by usingMATLAB. In each case, generate 100 test ma-trices and determine how many of the matricesare singular.

5. If a matrix is sensitive to roundoff errors, the com-puted value of its determinant may differ drasticallyfrom the exact value. For an example of this, set

U = round(100 ∗ rand(10));U = triu(U, 1) + 0.1 ∗ eye(10)

In theory,

det(U ) = det(U T ) = 10−10

and

det(UU T ) = det(U ) det(U T ) = 10−20

Compute det(U ), det(U ′), and det(U ∗ U ′) withMATLAB. Do the computed values match the the-oretical values?

6. Use MATLAB to construct a matrix A by setting

A = vander(1 : 6); A = A − diag(sum(A′))

(a) By construction, the entries in each row of Ashould all add up to zero. To check this, setx = ones(6, 1) and use MATLAB to com-pute the product Ax. The matrix A should besingular. Why? Explain. Use the MATLABfunctions det and inv to compute the valuesof det(A) and A−1. Which MATLAB functionis a more reliable indicator of singularity?

(b) Use MATLAB to compute det(AT ). Are thecomputed values of det(A) and det(AT ) equal?Another way to check if a matrix is singular isto compute its reduced row echelon form. UseMATLAB to compute the reduced row echelonforms of A and AT .


(c) To see what is going wrong, it helps to knowhow MATLAB computes determinants. TheMATLAB routine for determinants first com-putes a form of the LU factorization of the ma-trix. The determinant of the matrix L is ±1,depending on whether an even or odd numberof row interchanges were used in the compu-tation. The computed value of the determinantof A is the product of the diagonal entries ofU and det(L) = ±1. In the special case thatthe original matrix has integer entries, the exactdeterminant should take on an integer value.So in this case MATLAB will round its deci-mal answer to the nearest integer. To see whatis happening with our original matrix, use the

following commands to compute and displaythe factor U :

format short e[ L , U ] = lu(A); U

In exact arithmetic, U should be singu-lar. Is the computed matrix U singular? Ifnot, what goes wrong? Use the following com-mands to see the rest of the computation ofd = det(A):

format shortd = prod(diag(U ))

d = round(d)

CHAPTER TEST A True or False

For each of the statements that follow, answer true if thestatement is always true and false otherwise. In the caseof a true statement, explain or prove your answer. In thecase of a false statement, give an example to show thatthe statement is not always true. Assume that all thegiven matrices are n × n.

1. det(AB) = det(B A)

2. det(A + B) = det(A) + det(B)

3. det(cA) = c det(A)

4. det((AB)T ) = det(A) det(B)

5. det(A) = det(B) implies A = B.

6. det(Ak) = det(A)k

7. A triangular matrix is nonsingular if and only if itsdiagonal entries are all nonzero.

8. If x is a nonzero vector in Rn and Ax = 0, then

det(A) = 0.

9. If A and B are row equivalent matrices, then theirdeterminants are equal.

10. If A �= O , but Ak = O (where O denotes the zeromatrix) for some positive integer k, then A must besingular.

CHAPTER TEST B

1. Let A and B be 3 × 3 matrices with det(A) = 4and det(B) = 6, and let E be an elementary ma-trix of type I. Determine the value of each of thefollowing:(a) det( 1

2 A) (b) det(B−1 AT ) (c) det(E A2)

2. Let

A =⎧⎪⎪⎪⎪⎪⎩

x 1 11 x −1

−1 −1 x

⎫⎪⎪⎪⎪⎪⎭(a) Compute the value of det(A). (Your answer

should be a function of x .)(b) For what values of x will the matrix be singu-

lar? Explain.

3. Let

A =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎩1 1 1 11 2 3 41 3 6 101 4 10 20

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎭

(a) Compute the LU factorization of A.

(b) Use the LU factorization to determine thevalue of det(A).

4. If A is a nonsingular n × n matrix, show that AT Ais nonsingular and det(AT A) > 0.

5. Let A be an n × n matrix. Show that if B = S−1 ASfor some nonsingular matrix S, then det(B) =det(A).

6. Let A and B be n × n matrices and let C = AB.Use determinants to show that if either A or B issingular, then C must be singular.

7. Let A be an n × n matrix and let λ be a scalar.Show that

det(A − λI ) = 0

if and only if

Ax = λx for some x �= 0

Chapter Two Exercises 109

8. Let x and y be vectors in Rn , n > 1. Show that if

A = xyT , then det(A) = 0.

9. Let x and y be distinct vectors in Rn (i.e., x �= y),

and let A be an n × n matrix with the property that

Ax = Ay. Show that det(A) = 0.

10. Let A be a matrix with integer entries. If| det(A)| = 1, then what can you conclude aboutthe nature of the entries of A−1? Explain.

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