Question

# The 2 times 2 matrices A and B below are related to matrix C by the equation: C=3A-2B. Which of the following is matrix C. A=begin{bmatrix}3 & 5 -2 &

Matrices

The $$2 \times 2$$ matrices A and B below are related to matrix C by the equation: $$C=3A-2B$$. Which of the following is matrix C.
$$A=\begin{bmatrix}3 & 5 \\-2 & 1 \end{bmatrix} B=\begin{bmatrix}-4 & 5 \\2 & 1 \end{bmatrix}$$
$$\begin{bmatrix}-1 & 5 \\2 & 1 \end{bmatrix}$$
$$\begin{bmatrix}-18 & 5 \\10 & 1 \end{bmatrix}$$
$$\begin{bmatrix}18 & -5 \\-10 & -1 \end{bmatrix}$$
$$\begin{bmatrix}1 & -5 \\-2 & -1 \end{bmatrix}$$

2020-12-26
Step 1
We have to find matrix C by the equation C=3A−2B where matrices are given as:
$$A=\begin{bmatrix}3 & 5 \\-2 & 1 \end{bmatrix} \text{ and } B=\begin{bmatrix}-4 & 5 \\2 & 1 \end{bmatrix}$$
We know the operations of matrices,
If we multiply by any scalar to the matrix then it get multiplied in each elements example:
$$2\begin{bmatrix}a & b \\c & d \end{bmatrix}=\begin{bmatrix}2a & 2b \\2c & 2d \end{bmatrix}$$
$$\begin{bmatrix}a & b \\c & d \end{bmatrix}+\begin{bmatrix}x & y \\z & w \end{bmatrix}=\begin{bmatrix}a+x & b+y \\c+z & d+w \end{bmatrix}$$
Step 2
Applying above rule for the given condition, we get
$$C=3A-2B$$
$$=3\begin{bmatrix}3 & 5 \\-2 & 1 \end{bmatrix}-2\begin{bmatrix}-4 & 5 \\2 & 1 \end{bmatrix}$$
$$=\begin{bmatrix}3 \times 3 & 3\times5 \\3\times (-2) & 3\times 1 \end{bmatrix}-\begin{bmatrix}2\times (-4) & 2\times 5 \\2\times 2 & 2\times 1 \end{bmatrix}$$
$$=\begin{bmatrix}9 & 15 \\-6 & 3 \end{bmatrix}-\begin{bmatrix}-8 & 10 \\4 & 2 \end{bmatrix}$$
$$=\begin{bmatrix}9-(-8) & 15-10 \\-6-4 & 3-2 \end{bmatrix}$$
$$=\begin{bmatrix}9+8 & 5 \\-10 & 1 \end{bmatrix}$$
$$=\begin{bmatrix}17 & 5 \\-10 & 1 \end{bmatrix}$$
Hence, value of C is $$\begin{bmatrix}17 & 5 \\-10 & 1 \end{bmatrix}$$
Note:
There is no suitable option for the given conditions.