8.2

8.2Operations with

Matrices

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Matrix Addition and Scalar Multiplication

With matrix addition, you can add two matrices (of the same order) by adding their corresponding entries.

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Example 1 – Addition of Matrices

What has to be true about the

orders of 2 matrices to be able

to add or subtract them?

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Example 2 – Addition of Matrices

d. The sum of

and

is undefined because A is of order 3 3 and B is of

order 3 2.

cont’d

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In operations with matrices, numbers are usually referred to as scalars

You can multiply a matrix A by a scalar c by multiplying each entry in A by c.

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Example 3 – Scalar Multiplication and Matrix Subtraction

For the following matrices, find (a) 3A, (b) –B, and (c) 3A – B.

(a) Find 3A

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(b) Find -B

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(c) Find Find 3A-B

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Matrix Multiplication

For the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix.

That is, the middle two numbers must be the same. The outside two numbers give the order of the product, as shown below.

A B = AB

m n n p m p

The INNER dimensions of the two matrices have to be the same in order to multiply them!

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Example 4 – Finding the Product of Two Matrices

Find the product AB using and

Solution:

To find the entries of the product, multiply each row of A by each column of B

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Finish on Whiteboard

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The properties of matrix addition and scalar multiplication are similar to those of addition and multiplication of real numbers.

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if A is an m n matrix and O is the m n zero

matrix consisting entirely of zeros, then A + O = A.

O is the additive identity for the set of all m n matrices.

For example, the following matrices are the additive identities for the sets of all 2 3 and 2 2 matrices.

and2 3 zero matrix 2 2 zero matrix

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Example 4 – Solution cont’d

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Be sure you understand that for the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix.

That is, the middle two indices must be the same. The outside two indices give the order of the product, as shown below.

A B = ABm n n p m p

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Even if both AB and BA are defined, matrix multiplication is not, in general, commutative. That is, for most matrices, AB BA. This is one way in which the algebra of real numbers and the algebra of matrices differ.


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If A is an n n matrix, the identity matrix has the property that AI = A and I A = A. For example,

and

AI = A

IA = A

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8.2

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