3.5 Perform Basic Matrix Operations

Matrix

A rectangular arrangement of numbers in rows and columns

Dimensions

The dimensions of a matrix with m rows and n columns are m ´ n.

Elements

The numbers in a matrix

Equal matrices

Matrices that have the same dimensions and equal elements in corresponding positions

Scalar

A real number by which you multiply a matrix

Scalar multiplication

The process of multiplying each element in a matrix by a scalar

ADDING AND SUBTRACTING MATRICES

To add or subtract two matrices, simply add or subtract _corresponding_ elements. You can add or subtract matrices only if they have the same _dimensions_.

Adding Matrices
Subtracting Matrices

Example 1

Add and subtract matrices

Perform the indicated operation, if possible.

Solution

a. The dimensions of are _2 ´ 2_ and the dimensions of

are _2 ´ 1_. So, you _cannot_ subtract the matrices.

b. The dimensions are _the same . So, add the matrices.

Example 2

Multiply a matrix by a scalar

Perform the indicated operation, if possible.

a.

b.

PROPERTIES OF MATRIX OPERATIONS

Let A, B, and C be matrices with the same dimensions, and let k be a scalar.

Associative Property of Addition

(A + B) + C = _A + (B + C)_

Commutative Property of Addition A + B = _B + A_

Distributive Property of Addition k(A + B) = _kA + kB_

Distributive Property of Subtraction k(A - 6) = _kA - kB_

Example 3

Solve a multi-step problem

Pet Stores Two pet stores sell both dogs and cats. Sales from each store for last month and this month are shown below.

Last Month: Store 1 sold 42 dogs, 33 cats.

Store 2 sold 56 dogs, 21 cats.

This Month: Store 1 sold 36 dogs, 51 cats.

Store 2 sold 48 dogs, 37 cats.

Organize the data into matrices, then write and interpret a matrix giving the average monthly sales for the two month period.

Organize the data into two 2 ´ 2 matrices.

Last month (A) This month(B)

dogs cats dogs cats

Store 1 Store 1

Store 2 Store 2

Write a matrix for the average monthly sales by adding

A and B and then multiplying the result by

Store 1 sold an average of _39_ dogs and _42_ cats, while

Store 2 sold an average of _52_ dogs and _29_ cats.

Example 4

Solve a matrix equation

Solve the matrix equation for x and y.

Solution

Simplify the left side of the equation.

	Write original equation.
	Add matrices inside parentheses.