Section 3.1The Determinant of a Matrix

Determinants are computed only on square matrices.Notation: det(A) or |A| 

For 1x1 matrices:det( [k] ) = k

 

Determinants are computed only on square matrices.Notation: det(A) or |A| 

For 1x1 matrices:det( [k] ) = k

 

For 2x2 matrices:

 

For larger matrices, we define a determinant in terms of cofactors.

deta b

ad bcc d

Def. If A is a square matrix, then the ij minor, denoted Mij, is the determinant of the matrix obtained by deleting the ith row and the jth column of A.

The ij cofactor , denoted Cij, is given by Cij = (-1)i+jMij

Ex. Find C23 and C13 for the matrix

2 1 0

2 1 4

0 2 3

Computing determinants by the cofactor expansion.

The determinant of an nxn matrix A can be computed by expanding along the ith row:

The determinant of an nxn matrix A can be computed by expanding along the jth column:

1

detn

ik ikk

A a C

1

detn

kj kjk

A a C

Ex. Compute the determinant of

2 1 0

2 1 4

0 2 3

Ex. Compute the determinant of

1 2 3 0

1 1 0 2

0 2 0 3

3 4 0 1

Triangular matrices:

  

If A is a triangular matrix then det(A) = a11 a22 a33 · · · ann

3 4 9

0 1 4

0 0 11

8 0 0

4 10 0

1 2 2

2 3 0 1

0 1 55 10

0 0 4 99

0 0 0 3

2 0

5 17

Triangular matrices:

  

A =

det(A) =

2 3 0 1

0 1 55 10

0 0 4 99

0 0 0 3

Section 3.2Evaluation of a Determinant Using

Elementary Operations

We computed determinants by the “cofactor expansion method” in the previous section. We shall introduce a new method which involves placing a given matrix into triangular form via elementary row operations.  Why even bother with a second method for computing determinants if we already have one that works?

There are some problems in math that are theoretically simple but practically impossible. Think, for example, of a determinant of a 50x50 matrix. When computed by expanding by cofactors, this involves :

50 different 49x49 determinants. Each one of these 49x49 determinants requires 49 different 48x48 determinants.Each one of these 48x48 determinants requires 48 different 47x47 determinants. Each one of these . . . .

We end up with a total of 50∙49∙48∙47∙ ∙ ∙6∙5∙4∙3 different 2x2 determinants (this is about 1064 2x2 determinants that must be calculated).

Even if a computer could calculate one million 2x2 determinants per second, it would take about 1058 seconds (about 1050 years) to finish calculating our 50x50 determinant. (According to the big bang theory, the universe is only about 1010 years old.)

Order n Cofactor Expansion Row Reduction

Additions Multiplications Additions Multiplications

3 5 9 5 10

5 119 205 30 45

10 3,628,799 6,235,300 285 339

Suppose that B is the triangular matrix obtained from A through row operations. We need to exploit the relationship between det(B) and det(A). To do so, we must first see how each row operation affects the value of a determinant.

Theorem:Suppose that A* was obtained from A through a single elementary row operation.

i. If that operation was Ri ↔ Rj then we have: det(A*) = –det(A).

ii. If that operation was Ri + cRj → Ri then we have: det(A*) = det(A).

iii. If that operation was cRi → Ri then we have: det(A*) = c det(A).

Ex. Verify iii. above is true on the following matrices:

1 0 0

0 1 0

0 0 7

1 0 0

0 1 0

0 0 1

Ex. Suppose . Compute the determinants of the following matrices.

(a)

det 10

a b c

d e f

g h i

3 3 3

4 4 4

a b c

d e f

g a h b i c

Ex. Suppose . Compute the determinants of the following matrices.

(b)

det 10

a b c

d e f

g h i

2 2 2a b c

g h i

d e f

Ex. Suppose . Compute the determinants of the following matrices.

(c)

det 10

a b c

d e f

g h i

5 5 5

a b c

a b c

g h i

Ex. Use elementary row operations to compute the determinant of 2 3 10

1 2 2

4 9 11

If det(A) = 0, what do we know about the triangular matrix obtained by applying row operations on A?

If det(A) ≠ 0, then there is only one solution to a system represented by AX = B.

Section 3.3Properties of Determinants

Let and . Then AB is .

Compute: det(A) det(B) det(AB)

1 2 2

0 3 2

1 0 1

A

2 0 1

0 1 2

3 1 2

B

8 4 1

6 1 10

5 1 1

Theorem:Suppose A and B are nxn matrices and c is a scalar.

1. det(AB) = _________________

2. det(cA) = _________________

3. det(AT) = _________________

det(A) det(B)

cn det(A)

det(A)

Theorem: det(A-1) = ___________

Theorem: If A is invertible then det(A) ≠ 0.

Theorem: Let A be an nxn matrix. The following are equivalent:

1. A is invertible.

2. AX = B has a unique solution.

3. AX = O has only the trivial solution.

4. rref(A) = I.

5. det(A) ≠ 0.

Section 3.5Applications of Determinants

Cramer’s Rule:

Consider the following system of linear equations represented by the matrix equation AX = B:

a11x1 + a12x2 + · · · + a1nxn = b1

a21x1 + a22x2 + · · · + a2nxn = b2

a31x1 + a32x2 + · · · + a3nxn = b3

: : :an1x1 + an2x2 + · · · + annxn = bn

 

Now, think of A in terms of its column vectors: A = [ a1 a2 a3 · · · an ]Define A1 = [ b a2 a3 · · · an ]

A2 = [ a1 b a3 · · · an ] A3 = [ a1 a2 b · · · an ] : : : An = [ a1 a2 a3 · · · b ]

If det(A) ≠ 0 then there is a unique solution to AX = B which can be computed by:

31 2

1 2 3

det( ) det( )det( ) det( ), , , ,

det det det detn

n

A AA Ax x x x

A A A A

Ex. Use Cramer’s rule to solve the following: x – y + 3z = 22x + y – z = 5–x + y – 4z = –4