2.2Multiplying Matrices

Mr. Anderson

Pre Calculus

Falconer Central High School

Falconer, NY

Ordering a MatrixHi. I’m calling to

see if I could order a matrix,

please.Why certainly.

Ordering a Matrix

(I, j) denotes the element in the ith row and the jth column. Best to have an example of course.

1 0 4 53 −2 6 11−1 10 13 00 −5 7 11

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

3,2( )=

1,4( )=

4,2( )=

Scalar Multiple

A scalar is a constant (a number). The product of a scalar and a matrix is the result of

“distributing” the scalar through the matrix. Example

let k =4 and A=1 02 43 6

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Then kA =4 08 16

12 24

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Multiplying Matrices

First, multiplication of matrices is not commutative. Ie. ORDER MATTERS!! Second, for two matrices to be multiplied, the number

of columns in the first matrix must be equal to the number of rows in the second matrix.

In other words, matrices A and B can only be multiplied if A is an m x n matrix and B is an n x r matrix.

Multiplying Matrices (cont.)

In other words, matrices A and B can only be multiplied if A is an m x n matrix and B is an n x r matrix.

Furthermore, AB will be an m x r matrix. Ie. AB will have m rows and r columns. Officially: When finding the product AB of two matrices, the entry in

(i, j) of AB is the sum of the products of the corresponding elements in row i of matrix A and column j of matrix B.

Say what?

When finding the product AB of two matrices, the entry in (i, j) of AB is the sum of the products of the corresponding elements in row i of matrix A and column j of matrix B.

Example:

A = 4 35 2

⎡⎣⎢

⎤⎦⎥; B= 8 0

9 −6⎡⎣⎢

⎤⎦⎥

Find AB

Question: Does the number of columns in A equal the number of rows in B?

Example 2

A = 2 0 13 6 0

⎡⎣⎢

⎤⎦⎥

B=1 12 36 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Find AB

Example 3

A = 1 03 6

⎡⎣⎢

⎤⎦⎥

B= 2 −1 0 5 10 0 −1 3 1

⎡⎣⎢

⎤⎦⎥

Find AB

Homework

Pg. 69 # 29-40 [5]