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.
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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