Matrix Arithmetic: Addition, Subtraction, and Scalar Multiplication (VCE SSCE General Mathematics): Revision Notes

Worked Example: Finding values in equal matrices

Question: Given that $\begin{bmatrix} 2 & a \\ 8 & b+1 \end{bmatrix} = \begin{bmatrix} 2 & 10 \\ 20 & 7 \end{bmatrix}$, find the value of $a$ and the value of $b$.

Solution:

Since the matrices are equal, each corresponding element must be equal.

Comparing the elements in position (1,2): $a = 10$

Comparing the elements in position (2,1): $8 = 20$ (this would indicate an error, but let's check position (2,2))

Wait, looking at the correct positions:

• Position (2,1): $8 = 20$ doesn't work, so let me reconsider...

Actually, from the source:

• $a = 10$ (from comparing position (1,2))
• $b + 1 = 7$ which means $b = 6$ (from comparing position (2,2))

Note: The element in position (2,1) means both matrices should have 8 in that position, but the second matrix shows 20. Based on the worked solution provided, we have:

$a = 10$ and $b = 6$

Worked Example: Adding two matrices

Question: If $A = \begin{bmatrix} 2 & 3 & 0 \\ 1 & 4 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 & 3 \\ 2 & -2 & 1 \end{bmatrix}$, find $A + B$.

Solution:

Step 1: Check that both matrices have the same order.

Both matrices are $2 \times 3$ (2 rows, 3 columns), so they can be added.

Step 2: Add corresponding elements.

Worked Example: Subtracting two matrices

Question: If $A = \begin{bmatrix} 2 & 3 & 0 \\ 1 & 4 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 & 3 \\ 2 & -2 & 1 \end{bmatrix}$, find $A - B$.

Solution:

Step 1: Check that both matrices have the same order.

Both matrices are $2 \times 3$, so they can be subtracted.

Step 2: Subtract corresponding elements.

Worked Example: Scalar multiplication

Question: If $A = \begin{bmatrix} 2 & 3 & 0 \\ 1 & 4 & 2 \end{bmatrix}$ and $C = \begin{bmatrix} 4 & -4 \\ -2 & 6 \end{bmatrix}$, find $3A$ and $0.5C$.

Solution:

For $3A$, multiply each element in matrix $A$ by 3:

$3A = 3 \begin{bmatrix} 2 & 3 & 0 \\ 1 & 4 & 2 \end{bmatrix} = \begin{bmatrix} 6 & 9 & 0 \\ 3 & 12 & 6 \end{bmatrix}$

For $0.5C$, multiply each element in matrix $C$ by 0.5:

$0.5C = 0.5 \begin{bmatrix} 4 & -4 \\ -2 & 6 \end{bmatrix} = \begin{bmatrix} 2 & -2 \\ -1 & 3 \end{bmatrix}$

Worked Example: Proving a result equals the zero matrix

Question: If $G = \begin{bmatrix} 6 & 0 \\ -4 & 2 \end{bmatrix}$ and $H = \begin{bmatrix} 9 & 0 \\ -6 & 3 \end{bmatrix}$, show that $3G - 2H = O$.

Solution:

First, calculate $3G$:

$3G = 3 \times \begin{bmatrix} 6 & 0 \\ -4 & 2 \end{bmatrix} = \begin{bmatrix} 18 & 0 \\ -12 & 6 \end{bmatrix}$

Next, calculate $2H$:

$2H = 2 \times \begin{bmatrix} 9 & 0 \\ -6 & 3 \end{bmatrix} = \begin{bmatrix} 18 & 0 \\ -12 & 6 \end{bmatrix}$

Now subtract:

Therefore, $3G - 2H = O$

Worked Example: Processing sales data

Question: The sales data for two used car dealers, Honest Joe's and Super Deals, are displayed in the table below:

a) Construct two matrices, $A$ and $B$, to represent the sales data for 2014 and 2015 separately.

Solution:

$A = \begin{bmatrix} 24 & 32 & 11 \\ 32 & 34 & 9 \end{bmatrix} \text{ (2014 sales)}$

$B = \begin{bmatrix} 26 & 38 & 16 \\ 35 & 41 & 12 \end{bmatrix} \text{ (2015 sales)}$

b) Construct a new matrix $C = A + B$. What does this matrix represent?

Solution:

Matrix $C$ represents the total sales for 2014 and 2015 combined for the two dealers.

c) Construct a new matrix $D = B - A$. What does this matrix represent?

Solution:

Matrix $D$ represents the increase in sales from 2014 to 2015 for the two dealers.

d) Both dealers want to increase their 2015 sales by 50% by 2016. Construct a new matrix $E = 1.5B$. Explain why this matrix represents the planned sales figures for 2016.

Solution:

Multiplying by 1.5 (which equals $1 + 0.5$) increases each value by 50%. This is because $1.5 \times x = x + 0.5x$, representing 100% of the original plus an additional 50%.