45

2.1: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are 2 matrices equal? Scalar Multiplication

46

Matrix Addition

Theorem 1 (pg 130)

Let A, B, and C be matrices of the same size, and let r & s be scalars.

1.) ABBA +=+

2.) C)(BACB)(A ++=++

3.) A0A =+

4.) rBrAB)r(A +=+

5.) sArAs)A(r +=+

6.) (rs)Ar(sA) =

Example 1

Let

−=

01

32

46

A and

−

−

=

42

15

38

B . Compute each of the following:

=2A-

=2A-B

=+ 2BA

47

MATRIX MULTIPLICATION How do I multiply matrices? Row-Column Rule For Computing AB

Theorem 2 (pg 134)

Let A be an mxn matrix, and let B & C have sizes for which the indicated sums and products are defined. Let r be any scalar.

1.) (AB)CA(BC) =

2.) ACABC)A(B +=+

3.) CABAC)A(B +=+

4.) A(rB)(rA)Br(AB) ==

5.) nm AAA II ==

Associative law of multiplication

Left distributive law

Right distributive law

Identity for matrix multiplication

48

Example 2

Let

−

−=

254

102A ,

−=

12

21B , and

−=

41

53C .

Compute each of the following:

=AC

=CA

=BC

=CB

49

WARNINGS!

1.) In general, BAAB ≠ .

2.) The cancellation laws do NOT hold for matrix multiplication. That is, if ACAB = then it is

NOT necessarily true that CB = .

3.) If a product AB is the zero matrix, you CANNOT conclude in general that 0A = or 0B = .

What is the transpose of a matrix?

50

Theorem 3 (pg 136)

Let A and B denote matrices whose sizes are appropriate for the following sums and products and let r be any scalar.

1.) ( ) AATT =

2.) ( ) TTTBABA +=+

3.) ( ) TTrArA =

4.) ( ) TTTABAB =

Example 3

Let

=

43

21A and

=

6

5x . Compute each of the following:

( ) =TAx

=TTAx

=Txx

=xxT

Is TTxA defined? Why or why not.

51

2.2: THE INVERSE OF A MATRIX When is a matrix A invertible? Singular Matrix vs. Nonsingular Matrix Theorem 4 (CAUTION: THIS ONLY WORKS FOR 2x2 MATRICES!)

Example 1

Find the inverse (if it exists) of each of the following matrices.

34

68

86

42

52

Theorem 5 Theorem 6 Theorem 7

Algorithm for Finding 1A − (pg 153)

1.) Set up the augmented matrix [A I].

2.) Row reduce the matrix into reduced echelon form.

3.) If A is row equivalent to I, then [A I] is row equivalent to [I 1A − ]. Otherwise, A does not

have an inverse.

53

Example 2

Let

=

801

352

321

A and

=

5

12

42

b . Find 1A − and use it to solve bxA = .

54

Example 3

Let A, B, C, D, X, and Y be invertible nxn matrices. Solve the equation, ( ) YDCXBA =+ for X.

Things to keep in mind: Matrix division does not exist. You cannot divide by a matrix. Also, keep

the order of multiplication consistent. If you multiply by 1A − , on the left of the left side of the

equation you must multiply by 1A − on the left of the right side of the equation as well.

55

2.3: CHARACTERIZATIONS OF INVERTIBLE MATRICES

Theorem 8: The Invertible Matrix Theorem (pg 163)

Let A be a square nxn matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false.

1.) A is an invertible matrix.

2.) A is row equivalent to the nxn identity matrix, nI .

3.) A has n pivot positions.

4.) The equation 0xA = has only the trivial solution.

5.) The columns of A form a linearly independent set.

6.) The linear transformation nn RR:T → given by ( ) xAxT = is one-to-one.

7.) The equation bxA = has exactly one solution for each b in nR .

8.) The columns of A span nR .

9.) The linear transformation nn RR:T → given by ( ) xAxT = is onto.

10.) There is an nxn matrix C such that nACCA I== .

11.) TA is an invertible matrix.

56

What is an invertible transformation? Theorem 9

What if nn RR:T → is one-to-one? Onto?

57

Example 1

Let 33 RR:T → by

+−

−+

−

=

31

321

32

3

2

1

x4x6

xx3x2

x3x

x

x

x

T be a linear transformation. Show that T is

invertible and find 1T− .

2.4 SUBSPACES OF nR

58

What is a vector space? What is a subspace? What is the subspace test?

59

Example 1 Determine which of the following are subspaces of R2.

60

Example 2

Use the subspace test to determine if the following is a subspace of 3R .

∈

+

−

+

= Ry,x

y4x3

xy

y2x

H

61

Example 3

Use the subspace test to determine if the following is a subspace of 3R .

∈

−

+

= Rc,b,a

cb

a

cb

W 2

62

What is ColA? What is NulA? Theorem 10

63

Example 4

Let

−

−

−

−−

=

075

387

312

121

A .

• What is ColA?

• ColA is a subspace of kR , what is k in this example?

• What is NulA?

• NulA is a subspace of sR , what is s in this example?

64

What is a basis?

What is the standard basis for nR ? Theorem 11

Example 5

Let

−

−

−

−−

=

075

387

312

121

A . Find a basis for ColA and NulA.

65

Example 6 Determine which sets are bases for R2 or R3. Justify each answer.

3

1,

−1

1

2

1,

− 3

2,

2

3

1

1

1

,

3

2

1

,

0

1

0

3

2

1

,

3

1

2

,

15

7

8

66

2.5 DIMENSION & RANK What is the dimension of a subspace?

Example 1

Determine the dimension of the subspace H of 3R spanned by the vectors

=

2

2

1

v1 ,

=

1

2

3

v 2 ,

=

7

10

11

v 3 ,

=

4

6

7

v 4 .

67

What is the rank of a matrix? Theorem 12 (The Rank Theorem)

Example 2 Suppose a 3x5 matrix A has 3 pivot columns.

Is 3RColA = ?

Is 2RNulA = ?

Suppose a 4x7 matrix A has 3 pivot columns.

Is 3RColA = ?

What is the dimension of NulA?

68

Example 3 Construct a 4x3 matrix with rank 1.

Theorem 13 (The Basis Theorem)

69

Example 4

Let

−

−−

−

−

−

=

42113

52221

14132

32310

13121

A . Is the set

−

=

2

1

3

3

0

,

1

1

2

0

1

,

9

0

9

3

6

S a basis for ColA?

70

Theorem 8: The Invertible Matrix Theorem (pg 190)

Let A be a square nxn matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false. 1.) A is an invertible matrix.

2.) A is row equivalent to the nxn identity matrix, nI .

3.) A has n pivot positions.

4.) The equation 0xA = has only the trivial solution.

5.) The columns of A form a linearly independent set.

6.) The linear transformation nn RR:T → given by ( ) xAxT = is one-to-one.

7.) The equation bxA = has exactly one solution for each b in nR .

8.) The columns of A span nR .

9.) The linear transformation nn RR:T → given by ( ) xAxT = is onto.

10.) There is an nxn matrix C such that nACCA I== .

11.) TA is an invertible matrix.

12.) The columns of A form a basis for nR .

13.) nRColA =

14.) ( ) nColAdim =

15.) ( ) nArank =

16.) }0{NulA =

17.) ( ) 0NulAdim =

18.) 0Adet ≠