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