Lesson Objectives

- Learn how to multiply a matrix by a scalar

## How to Multiply a Matrix by a Scalar

In this lesson, we will learn how to multiply a matrix by a scalar. When we talk about multiplying matrices, there are two scenarios. The first scenario, which is much simpler, involves multiplying some real number, called a scalar by a given matrix. The second scenario is where we will multiply two matrices together. This second case is much more involved, and we will look at that process in the next lesson. How do we multiply a matrix by a scalar? When we say a scalar, this just means some real number or to be more specific, a real number that is not inside of a matrix. If we take that scalar and multiply it by a matrix, this is where we get the term scalar multiplication.

If A=[a

So this is just very fancy notation that tells us to multiply our scalar or real number k by each and every element of our original matrix A to produce the matrix kA, which again is a scalar multiple of A. This is a very easy process, don’t let the notation scare you. Let’s look at an example.

Example #1: Find 2A. $$A=\left[ \begin{array}{cc}9&4\\ 6&2\end{array}\right] $$ Multiply each element of matrix A by 2: $$2A=\left[ \begin{array}{cc}18&8\\ 12&4\end{array}\right]$$

If A=[a

_{ij}] (again this lowercase a_{ij}is just generic notation to identify each individual element of our matrix A) is an m x n matrix and k is a scalar, the scalar multiple of A by k is the m x n matrix given by: kA=[ka_{ij}].So this is just very fancy notation that tells us to multiply our scalar or real number k by each and every element of our original matrix A to produce the matrix kA, which again is a scalar multiple of A. This is a very easy process, don’t let the notation scare you. Let’s look at an example.

Example #1: Find 2A. $$A=\left[ \begin{array}{cc}9&4\\ 6&2\end{array}\right] $$ Multiply each element of matrix A by 2: $$2A=\left[ \begin{array}{cc}18&8\\ 12&4\end{array}\right]$$

#### Skills Check:

Example #1

Find 2A. $$A=\left[ \begin{array}{cc}-3&5\\ 6&7\end{array}\right] $$

Please choose the best answer.

A

$$2A=\left[ \begin{array}{cc}2&5\\ 7&15\end{array}\right] $$

B

$$2A=\left[ \begin{array}{cc}6&8\\ 4&3\end{array}\right] $$

C

$$2A=\left[ \begin{array}{cc}-1&5\\ 6&11\end{array}\right] $$

D

$$2A=\left[ \begin{array}{cc}8&7\\ 4&6\end{array}\right] $$

E

$$2A=\left[ \begin{array}{cc}-6&10\\ 12&14\end{array}\right] $$

Example #2

Find 3A. $$A=\left[ \begin{array}{cc}3&6\\ 4&7\end{array}\right] $$

Please choose the best answer.

A

$$3A=\left[ \begin{array}{cc}-4&8\\ 2&1\end{array}\right] $$

B

$$3A=\left[ \begin{array}{cc}1&6\\ -4&8\end{array}\right] $$

C

$$3A=\left[ \begin{array}{cc}2&-1\\ -3&-5\end{array}\right] $$

D

$$3A=\left[ \begin{array}{cc}9&18\\ 12&21\end{array}\right] $$

E

$$3A=\left[ \begin{array}{cc}-4&-3\\ 9&15\end{array}\right] $$

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