What must be the matrix if $2X + \left[ {\begin{array}{*{20}{c}}
  1&2 \\
  3&4
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  3&8 \\
  7&2
\end{array}} \right]$
Option:
A. $\left[ {\begin{array}{*{20}{c}}
  1&3 \\
  2&{ - 1}
\end{array}} \right]$
B. $\left[ {\begin{array}{*{20}{c}}
  1&{ - 3} \\
  2&{ - 1}
\end{array}} \right]$
C. $\left[ {\begin{array}{*{20}{c}}
  2&6 \\
  4&{ - 2}
\end{array}} \right]$
D. $\left[ {\begin{array}{*{20}{c}}
  2&{ - 6} \\
  4&{ - 2}
\end{array}} \right]$

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 $2X + \left[ {\begin{array}{*{20}{c}}
  1&2 \\
  3&4
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  3&8 \\
  7&2
\end{array}} \right]$ .
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:
$2X = \left[ {\begin{array}{*{20}{c}}
  3&8 \\
  7&2
\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}
  1&2 \\
  3&4
\end{array}} \right]$
We get
$2X = \left[ {\begin{array}{*{20}{c}}
  2&6 \\
  4&{ - 2}
\end{array}} \right]$
$ \Rightarrow X = \dfrac{1}{2}\left[ {\begin{array}{*{20}{c}}
  2&6 \\
  4&{ - 2}
\end{array}} \right]$
On further evaluating, we get matrix
$ \Rightarrow X = \left[ {\begin{array}{*{20}{c}}
  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.