Mathh 1300: Section 4- 4 Matrices: Basic Operations

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Addition and SubtractionProduct of a Number k and a Matrix M

The Matrix Product

Math 1300 Finite MathematicsSection 4.4 Matrices: Basic Operations

Jason Aubrey

Department of MathematicsUniversity of Missouri

June 20, 2011

Jason Aubrey Math 1300 Finite Mathematics

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The Matrix Product

Two matrices are equal if they have the same size andtheir corresponding elements are equal.

For example,1 3 24 −1 −21 0 −1

1 3 24 −1 −21 0 −1

The sum of two matrices of the same size is the matrixwith elements that are the sum of the correspondingelements of the two given matrices.Addition is not defined for matricies of different sizes.

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The Matrix Product

Two matrices are equal if they have the same size andtheir corresponding elements are equal. For example,1 3 2

4 −1 −21 0 −1

1 3 24 −1 −21 0 −1

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The Matrix Product

4 −1 −21 0 −1

1 3 24 −1 −21 0 −1

The sum of two matrices of the same size is the matrixwith elements that are the sum of the correspondingelements of the two given matrices.

Addition is not defined for matricies of different sizes.

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The Matrix Product

4 −1 −21 0 −1

1 3 24 −1 −21 0 −1

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The Matrix Product

Example: Evaluate1 1 24 −3 −21 0 −4

−2 3 24 −1 −20 0 −4

−1 4 48 −4 −41 0 −8

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The Matrix Product

Example: Evaluate1 1 24 −3 −21 0 −4

−2 3 24 −1 −20 0 −4

−1 4 48 −4 −41 0 −8

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The Matrix Product

Example: Evaluate

[5 0 −21 −3 8

−1 70 6−2 8

Undefined!

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The Matrix Product

Example: Evaluate

[5 0 −21 −3 8

−1 70 6−2 8

Undefined!

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The Matrix Product

If A, B, and C are matrices of the same size, then

A + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)

A matrix with elements that are all zeros is called a zeromatrix.

For example,[0 00 0

0 0 00 0 00 0 0

, etc.

The negative of a matrix M, denoted −M is a matrix withelements that are the negatives of the elements in M. For

example, if M =

[1 −3 22 −4 3

]then −M =

[−1 3 −2−2 4 −3

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The Matrix Product

If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)

(A + B) + C = A + (B + C) (Associativity)

For example,[0 00 0

0 0 00 0 00 0 0

, etc.

example, if M =

[1 −3 22 −4 3

]then −M =

[−1 3 −2−2 4 −3

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The Matrix Product

If A, B, and C are matrices of the same size, thenA + B = B + A (Commutativity)(A + B) + C = A + (B + C) (Associativity)

For example,[0 00 0

0 0 00 0 00 0 0

, etc.

example, if M =

[1 −3 22 −4 3

]then −M =

[−1 3 −2−2 4 −3

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The Matrix Product

For example,[0 00 0

0 0 00 0 00 0 0

, etc.

example, if M =

[1 −3 22 −4 3

]then −M =

[−1 3 −2−2 4 −3

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The Matrix Product

For example,[0 00 0

0 0 00 0 00 0 0

, etc.

example, if M =

[1 −3 22 −4 3

]then −M =

[−1 3 −2−2 4 −3

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The Matrix Product

For example,[0 00 0

0 0 00 0 00 0 0

, etc.

The negative of a matrix M, denoted −M is a matrix withelements that are the negatives of the elements in M.

example, if M =

[1 −3 22 −4 3

]then −M =

[−1 3 −2−2 4 −3

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The Matrix Product

For example,[0 00 0

0 0 00 0 00 0 0

, etc.

example, if M =

[1 −3 22 −4 3

]then −M =

[−1 3 −2−2 4 −3

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The Matrix Product

If A and B are matrices of the same size, we define subtractionas follows:

A − B = A + (−B)

For example,

[3 1 2−3 3 1

2 2 −3−1 1 0

[1 −1 5−2 2 1

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The Matrix Product

If A and B are matrices of the same size, we define subtractionas follows:

A − B = A + (−B)

For example,

[3 1 2−3 3 1

2 2 −3−1 1 0

[1 −1 5−2 2 1

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The Matrix Product

The product of a number k and a matrix M, denoted kM, is amatrix formed by multiplying each element of M by k .

For example,

−1 04 3−2 1/3

−3 012 9−6 1

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The Matrix Product

The product of a number k and a matrix M, denoted kM, is amatrix formed by multiplying each element of M by k .

For example,

−1 04 3−2 1/3

−3 012 9−6 1

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The Matrix Product

Examples: Perform the indicated operations, if possible, giventhat

1 41 33 −7

and B =

0 1−2 24 1

3A + B

3 123 99 −21

0 1−2 24 1

3 131 11

13 −20

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The Matrix Product

1 41 33 −7

and B =

0 1−2 24 1

3A + B 3 123 99 −21

0 1−2 24 1

3 131 11

13 −20

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The Matrix Product

1 41 33 −7

and B =

0 1−2 24 1

3A + B 3 123 99 −21

0 1−2 24 1

3 131 11

13 −20

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The Matrix Product

A − B

1 41 33 −7

0 −12 −2−4 −1

1 33 1−1 −8

−A + 2B

−1 −4−1 −3−3 7

0 2−4 48 2

−1 −2−5 15 9

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The Matrix Product

A − B 1 41 33 −7

0 −12 −2−4 −1

1 33 1−1 −8

−A + 2B

−1 −4−1 −3−3 7

0 2−4 48 2

−1 −2−5 15 9

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The Matrix Product

A − B 1 41 33 −7

0 −12 −2−4 −1

1 33 1−1 −8

−A + 2B

−1 −4−1 −3−3 7

0 2−4 48 2

−1 −2−5 15 9

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The Matrix Product

A − B 1 41 33 −7

0 −12 −2−4 −1

1 33 1−1 −8

−A + 2B −1 −4

−1 −3−3 7

0 2−4 48 2

−1 −2−5 15 9

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The Matrix Product

A − B 1 41 33 −7

0 −12 −2−4 −1

1 33 1−1 −8

−A + 2B −1 −4

−1 −3−3 7

0 2−4 48 2

−1 −2−5 15 9

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The Matrix Product

DefinitionThe product of a 1× n row matrix and an n × 1 column matrix isa 1 × 1 matrix given by:

[a1 a2 . . . an]

b1b2...

= [a1b1 + a2b2 + · · ·+ anbn]

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The Matrix Product

DefinitionThe product of a 1× n row matrix and an n × 1 column matrix isa 1 × 1 matrix given by:

[a1 a2 . . . an]

b1b2...

= [a1b1 + a2b2 + · · ·+ anbn]

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The Matrix Product

Example:

[1 2 −2

= [1(0) + 2(2)− 2(1)] = [2]

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The Matrix Product

Example:

[1 2 −2

= [1(0) + 2(2)− 2(1)]

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The Matrix Product

Example:

[1 2 −2

= [1(0) + 2(2)− 2(1)] = [2]

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The Matrix Product

Definition (General Matrix Product)If A is an m × p matrix and B is a p × n matrix, the matrixproduct of A and B, denoted AB, is an m × n matrix whoseelements in the i th row and j th column is the real numberobtained from the product of the i th row of A and the j th columnof B. If the number of columns in A does not equal the numberof rows in B, the matrix product AB is not defined.

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The Matrix Product

The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size

3 × 3. However, the product BA has size 1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3. However, the product BA is undefined.

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The Matrix Product

The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3.

However, the product BA has size 1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3. However, the product BA is undefined.

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The Matrix Product

The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3. However, the product BA has size

1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3. However, the product BA is undefined.

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The Matrix Product

The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3. However, the product BA has size 1 × 1.

The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3. However, the product BA is undefined.

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The Matrix Product

The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3. However, the product BA has size 1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size

2 × 3. However, the product BA is undefined.

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The Matrix Product

The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3. However, the product BA has size 1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3.

However, the product BA is undefined.

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The Matrix Product

The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3. However, the product BA has size 1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3. However, the product BA

is undefined.

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The Matrix Product

The product AB of a 3 × 1 matrix A and a 1 × 3 matrix Bhas size 3 × 3. However, the product BA has size 1 × 1.The product AB of a 2 × 4 matrix A and a 4 × 3 matrix Bhas size 2 × 3. However, the product BA is undefined.

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals

3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0

− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 73 − 3 96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals

3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0

− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 73 − 3 96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8)

3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0

− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 73 − 3 96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8)

3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0

− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13

1 73 − 3 96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2)

3(2) + 1(3)− 2(1)3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0

− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13

1 73 − 3 96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2)

3(2) + 1(3)− 2(1)3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0

− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1

73 − 3 96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1

73 − 3 96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 7

3 − 3 96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0

3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 7

3 − 3 96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0

3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 73

− 3 96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0 3(−1) + 0 + 0

3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 73

− 3 96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0 3(−1) + 0 + 0

3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 73 − 3

96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0

− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 73 − 3

96 0 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0

− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 73 − 3 9

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8)

− 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 73 − 3 9

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8)

− 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 73 − 3 96

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2)

− 2(2) + 1(3) + 1(1)

− 13 1 73 − 3 96

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2)

− 2(2) + 1(3) + 1(1)

− 13 1 73 − 3 96 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 73 − 3 96 0

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The Matrix Product

Example: Find 3 1 −23 1 0−2 1 1

1 −1 20 0 38 −2 1

This equals 3(1) + 0 − 2(8) 3(−1) + 0 − 2(−2) 3(2) + 1(3)− 2(1)

3(1) + 0 + 0 3(−1) + 0 + 0 3(2) + 1(3) + 0− 2(1) + 0 + 1(8) − 2(−1) + 0 + 1(−2) − 2(2) + 1(3) + 1(1)

− 13 1 73 − 3 96 0 0

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