12.2 Multiplication of Matrices

Matrix Multiplication

The product of two matrices, Am×p and Bp×n, is the matrix AB with dimensions m × n. Any element in the ith row and jth column of this product matrix is the sum of the products of the corresponding elements of the ith row of A and the jth column of B.

When you multiply matrices, they need to be conformable for multiplication. This means: # of columns in 1st matrix = # of rows in 2nd matrixEx 1) 4 5

2 37 2 and

5 61 3

A B

3 × 2 2 × 2match

dimensions of product3 × 2

To get each element:

this is the first row, first column so we take 1st row of A × 1st column of B

(4)(2) + (5)(5) = 8 + 25 = 33

d d

d d

d d

write yourself how to get this element

33 42

24 33

17 21

AB

(4)(3) + (5)(6) = 12 + 30 = 42(7)(2) + (2)(5) = 14 + 10 = 24

Ex 2) Find A2 (same A from Ex 1)

2

4 5 4 5

7 2 7 2

1 3 1 3

A

3 × 2 3 × 2

Wait!You can’t!So… undefined

We can solve for unknown elements in a matrix equation.

Ex 3) Solve for x and y. 3 1 4 2 3

2 0 4 5 4 8

x

y

3x – 4 = 2 3x = 6 x = 2

12 + 5y = –3 5y = –15 y = –3

The Identity MatrixThe identity matrix is the equivalent to the algebraic 1.Multiplying by it does not change the original.

2 2 3 3

1 0 01 0

0 1 00 1

0 0 1

I I

etc.

Pattern: 1’s along the diagonal & 0’s everywhere else

*If the product of two matrices is I, then they are inverses of each other.

You can also multiply by a 0 matrix to get an O matrix.

Properties of Matrix Multiplication for Square MatricesIf A, B, and C are n × n matrices, then AB is an n × n matrix. Closure (AB)C = A(BC) Associative In×nA = AIn×n = A Multiplicative Identity

On×nA = AOn×n = On×n Multiplicative Property of the Zero Matrix

A–1 is the multiplicative inverse of A if A–1 is defined and AA–1 = A–1A= In×n

Multiplicative Inverse

A(B + C)= AB + AC (B + C)A = BA + CA

Distributive Properties

What properties are not here?? Commutative!

When we “store” information in matrices, we may have to transpose them(switch rows & columns) to make them conformable for multiplication.Ex:

Boys 18 20 Per 3 18 17

Girls 17 14 Per 4 20 14tA A

Per 3 Per 4 Boys Girls

It’s still the same!

Ex 4) A fruit stand owner packages fruit in three different ways for gift packages. Economy package, E, has 6 apples, 3 oranges and 3 pears. Standard package, S, has 5 apples, 4 oranges and 4 pears. Luxury package, L, has 6 types of each fruit. The costs are $0.50 for an apple, $1.10 for an orange and $0.80 for a pear. What is the total cost of preparing each package of fruit?

E 6 3 3

cost $0.50 $1.10 $0.80 S 5 4 4

L 6 6 6

A B

Costapple orange pear

Number of Itemsapple orange pear

If we multiply in this state… the labels don’t match upcost per fruit package per fruit

cost per fruit fruit per package

6 5 6

$0.50 $1.10 $0.80 3 4 6 cost $8.70 $10.10 $14.40

3 4 6

tAB

PackageE S L

1 × 3 3 × 31 × 3

Homework

#1202 Pg 608 #1, 3, 5, 8, 15, 16, 18, 19, 20, 29, 31, 34, 41, 42