VM - Scalar Multiplication Lesson

Scalar Multiplication

Scalar multiplication just means to multiply a matrix by a constant.

For instance, let  $A=\begin{bmatrix} 1 & -2 & 5\\ 0 & 4 & -1 \end{bmatrix}$ then if we want to find 2A, we would multiply each entry by 2.

So,  $2A=\begin{bmatrix} 2 & -4 & 10\\ 0 & 8 & -2 \end{bmatrix}$

Try a few problems to see if you've got it:

$A=\begin{bmatrix} -4 & 7 & 9\\ 0 & 3 & 2 \end{bmatrix} B= \begin{bmatrix} 3 & 6 & 8\\ -1 & -2 & 0 \end{bmatrix} C=\begin{bmatrix} 1 & 0 & 5\\ 4 & 2 & -2\\ 7 & -3 & 1 \end{bmatrix} D= \begin{bmatrix} 7 & -2 & 0\\ 5 & 3 & 3\\ 6 & -1 & 4 \end{bmatrix}$

Try these problems to see if you get the correct solution.

1. Problem: -3B

• Solution: $-3B=\begin{bmatrix} -9& -18 & -24\\ 3 & 6 & 0 \end{bmatrix}$

2. Problem: (1/2)C

• Solution: $\frac{1}{2}C=\begin{bmatrix} \frac{1}{2}& 0 & \frac{5}{2}\\ 2 & 1 & -1\\ \frac{7}{2} & -\frac{3}{2} & \frac{1}{2} \end{bmatrix}$

3. Problem: 3C+2D

• Solution: $3C+2D=\begin{bmatrix} 17& -4 & 15\\ 22 & 12 & 0 \\ 33 & -11 & 11 \end{bmatrix}$

4. Problem: 4B - A

• Solution: $4B-A=\begin{bmatrix} 16 & 17 & 23\\ -4 & -11 & -2 \end{bmatrix}$

Application:

The revenue generated by sales in the Vancouver and Quebec branches of the A-Plus auto parts store was as follows:

Revenue generated by sales in the Vancouver and Quebec branches of the A-Plus auto parts store

	Vancouver	Quebec
Wiper Blades	140	150
Clean Fluid (bottles)	30	36
Floor Mats	96	48

If the Canadian dollar was worth $0.65 U.S. at the time, compute the revenue in U.S. dollars.

We need to multiply each revenue figure by 0.65. Let A be the matrix of revenue figures in Canadian dollars:

The revenue figures in U.S. dollars are then given by the scalar multiple

In other words, in U.S. dollars, $91 worth of wiper blades was sold in Vancouver, $68.25 worth of wiper blades was sold in Quebec, and so on.

Another example:

The A-Plus auto parts store had the following sales in its Vancouver store:

Vancouver Store Sales

	Vancouver
Wiper Blades	20
Clean Fluid (bottles)	10
Floor Mats	8

We need to multiply each sales figure by the corresponding price and then add the resulting revenue figures. We represent the sales by a column vector, as suggested by the table.

We put the selling prices in a row vector.

We can now compute the total revenue as the product

So, the sale of these items generated a total revenue of $266.00.

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