Question

What must be the matrix if $2X+\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}3& 8\\ 7& 2\end{array}\right]$
Option:
A. $\left[\begin{array}{cc}1& 3\\ 2& -1\end{array}\right]$
B. $\left[\begin{array}{cc}1& -3\\ 2& -1\end{array}\right]$
C. $\left[\begin{array}{cc}2& 6\\ 4& -2\end{array}\right]$
D. $\left[\begin{array}{cc}2& -6\\ 4& -2\end{array}\right]$

Answer 1

Hint: We will be using the concept of matrix subtraction to solve the equation. For the matrix subtraction, there should be an equal number of rows and columns. A null matrix is produced when a matrix is subtracted from itself, or when $A-A=0$ . Matrix subtraction is the addition of a matrix's negative to another matrix, i.e., $A-B=A+(-B)$ .
Formula Used: The difference between two matrices, $A$ and $B$ or $A-B$ is defined as: if there are two matrices, $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ of the same order $m\times n$ then:
$D=\left[d_{ij}\right]$ and $A-B=\left[a_{ij}\right]-\left[b_{ij}\right]$
$\because$ $\left[d_{ij}\right]=\left[a_{ij}\right]-\left[b_{ij}\right]$

Complete step by step solution: We have $2X+\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}3& 8\\ 7& 2\end{array}\right]$ .
To determine the value of the matrix $X$ , we have to solve the equation. To solve the equation, subtract both the given matrices as shown below:
$2X=\left[\begin{array}{cc}3& 8\\ 7& 2\end{array}\right]-\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$
We get
$2X=\left[\begin{array}{cc}2& 6\\ 4& -2\end{array}\right]$
$\Rightarrow X={\displaystyle \frac{1}{2}}\left[\begin{array}{cc}2& 6\\ 4& -2\end{array}\right]$
On further evaluating, we get matrix
$\Rightarrow X=\left[\begin{array}{cc}1& 3\\ 2& -1\end{array}\right]$ .

Option ‘A’ is correct

Note: Only when the two matrices are in the same order do they get subtracted. The two matrices cannot be subtracted from one another if the order is different. Because the 3 x 3 and 2 x 2 matrices have different orders or dimensions, it would not be possible to subtract one from the other. The order of the two matrices must match in order to subtract them.