Question

If $A=\left( \begin{matrix}
   1 & -2 \\
   3 & 0 \\
\end{matrix} \right)$ , $B=\left( \begin{matrix}
   -1 & 4 \\
   2 & 3 \\
\end{matrix} \right)$ , $C=\left( \begin{matrix}
   0 & -1 \\
   1 & 0 \\
\end{matrix} \right)$ then 5A-3B-2C is equal to ?
A . $\left( \begin{matrix}
   8 & 20 \\
   7 & 9 \\
\end{matrix} \right)$
B. $\left( \begin{matrix}
   8 & -20 \\
   7 & -9 \\
\end{matrix} \right)$
C. $\left( \begin{matrix}
   -8 & 20 \\
   -7 & 9 \\
\end{matrix} \right)$
D. $\left( \begin{matrix}
   8 & 7 \\
   -20 & -9 \\
\end{matrix} \right)$

Answer 1

Hint:In this question, we are given the matrix A, B, and C and we have to find the value of 5A-3B-2C. To solve this, first, we multiply 5 with matrix A, then 3 with matrix B, and 2 with matrix C. After multiplying them, we add and subtract the terms and get the desired result and choose the correct option.

Complete step by step Solution:
Given $A=\left( \begin{matrix}
   1 & -2 \\
   3 & 0 \\
\end{matrix} \right)$ , $B=\left( \begin{matrix}
   -1 & 4 \\
   2 & 3 \\
\end{matrix} \right)$ and $C=\left( \begin{matrix}
   0 & -1 \\
   1 & 0 \\
\end{matrix} \right)$
All the matrices are of $2\times 2$ order.
We have to find the value of 5A-3B-2C
First, we multiply 5 with matrix A
5A = 5$\left( \begin{matrix}
   1 & -2 \\
   3 & 0 \\
\end{matrix} \right)$
5A = $\left( \begin{matrix}
   5 & -10 \\
   15 & 0 \\
\end{matrix} \right)$
Now we multiply 3 with matrix B
3B = 3$\left( \begin{matrix}
   -1 & 4 \\
   2 & 3 \\
\end{matrix} \right)$
3B = $\left( \begin{matrix}
   -3 & 12 \\
   6 & 9 \\
\end{matrix} \right)$
Now we multiply 2 with matrix C
2C = 2$\left( \begin{matrix}
   0 & -1 \\
   1 & 0 \\
\end{matrix} \right)$
2C = $\left( \begin{matrix}
   0 & -2 \\
   2 & 0 \\
\end{matrix} \right)$
Now we add and subtract the matrices according to our given equation.
Now 5A-3B-2C = $\left( \begin{matrix}
   5 & -10 \\
   15 & 0 \\
\end{matrix} \right)$ - $\left( \begin{matrix}
   -3 & 12 \\
   6 & 9 \\
\end{matrix} \right)$ - $\left( \begin{matrix}
   0 & -2 \\
   2 & 0 \\
\end{matrix} \right)$
5A-3B-2C = $\left( \begin{matrix}
   5-(-3)-0 & -10-12-(-2) \\
   15-6-2 & 0-9-0 \\
\end{matrix} \right)$
5A-3B-2C = $\left( \begin{matrix}
   5+3 & -10-12+2 \\
   15-6-2 & 0-9-0 \\
\end{matrix} \right)$
Simplifying further, we get
5A-3B-2C = $\left( \begin{matrix}
   8 & -20 \\
   7 & -9 \\
\end{matrix} \right)$
Thus the value of 5A-3B-2C = $\left( \begin{matrix}
   8 & -20 \\
   7 & -9 \\
\end{matrix} \right)$

Therefore, the correct option is (B).

Note: Keep in mind before adding and subtracting any matrices that they have an equal number of columns and rows to be added. As the given matrices are of order $2\times 2$, so we can add it simply. Similarly we can add a $2\times 3$ matrix with a $2\times 3$ matrix or $3\times 3$ matrix with $3\times 3$ matrix. However, we cannot add $2\times 3$ matrix with a $3\times 2$ matrix. Similarly, we cannot add $2\times 2$ matrix with a $3\times 3$ matrix. The order in which we add the matrix is not important because the addition of two matrices is commutative.