# Scalar Multiplication: Product of a Scalar and a Matrix

Matrix multiplication usually falls into one of two types or classifications. The first one is called **Scalar Multiplication**, also known as the “**Easy Type**“; where you simply multiply a

The second one is called Matrix Multiplication which is discussed in a separate lesson.

In this lesson, we will focus on the “Easy Type” because the approach is extremely simple or straightforward.

## “Formula” of Scalar Multiplication (Easy Type)

Here’s the simple procedure as shown by the formula above.

**Take the number outside the matrix (known as the scalar) and multiply it by each and every entry or element of the matrix.**

### Examples of Scalar Multiplication

**Directions**: Given the following matrices, perform the indicated operation. Apply **scalar multiplication** as part of the overall simplification process.

**Example 1**: Perform the indicated operation for [latex]2A[/latex].

I will take the scalar 2 (similar to the coefficient of a term) and distribute it by multiplying it to each entry of matrix [latex]A[/latex]. In case you forgot, you may review the general formula above.

Since matrix [latex]A[/latex] is

then [latex]2A[/latex] is solved by…

That’s all there is to it. Done!

**Example 2**: Perform the indicated operation for [latex]-3B[/latex].

I will do the same thing similar to Example 1. No big deal! Multiply the negative scalar, [latex]−3[/latex], into each element of matrix [latex]B[/latex].

Since matrix [latex]B[/latex] is

then matrix [latex]-3B[/latex]** **is solved by…

Did you arrive at the same final answer? If not, please recheck your work to make sure that it matches the correct answer.

**Example 3**: Perform the indicated operation for [latex]-2D + 5F[/latex].

To solve this problem, I need to apply scalar multiplication twice and then add their results to get the final answer.

- First, find the value of matrix [latex]-2D[/latex]

I know that matrix [latex]D[/latex] is

Therefore, [latex]-2D[/latex] is obtained as follows using scalar multiplication.

- Second, find the value of [latex]5F[/latex]

Matrix [latex]F[/latex] is given as

That means [latex]5F[/latex] is solved using scalar multiplication.

- Now, I can solve for [latex]-2D+5F[/latex] by adding the values of matrices [latex]-2D[/latex] and [latex]5F[/latex], as shown above. Check out the solution to review how to add and subtract matrices.

That’s it!

**Example 4**: What is the difference of [latex]4A[/latex] and [latex]3C[/latex]?

At this point, you should have mastered already the skill of scalar multiplication. The very first step is to find the values of [latex]4A[/latex] and [latex]3C[/latex], respectively. Then we subtract the newly formed matrices, that is, [latex]4A-3C[/latex].

Finding the value of [latex]4A[/latex]

Finding the value of [latex]3C[/latex]

Finally, we subtract [latex]4A[/latex] by [latex]3C[/latex].

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