Question

Inverse matrix of $\left[ \begin{matrix} 4 & 7 \\ 1 & 2 \\ \end{matrix} \right]$ [RPET 1996, 2001]
A. $\left[ \begin{matrix} 2 & -7 \\ -1 & 4 \\ \end{matrix} \right]$
B. $\left[ \begin{matrix} 2 & -1 \\ -7 & 4 \\ \end{matrix} \right]$
C. $\left[ \begin{matrix} -2 & 7 \\ 1 & -4 \\ \end{matrix} \right]$
D. $\left[ \begin{matrix} -2 & 1 \\ 7 & -4 \\ \end{matrix} \right]$

Answer 1

Hint:
Using the determinant and adjoint of the given matrix you can determine the inverse of the matrix.


Formula Used:
Inverse matrix formula $A^{-1}=\dfrac{adjA}{|A|}$

Complete step-by-step solution:
Let $A=\left[ \begin{matrix} 4 & 7 \\ 1 & 2 \\ \end{matrix} \right]$
Determinant;
$|A|=(4\times2)-(7\times1)\\
|A|=8-7\\
|A|=1$
Now, the Adjoint of A can be determined by alternating the main diagonal components. Simply swap the signs of the components in the other diagonal, taking care not to interchange them.
Now, Adjoint of $A=\left[ \begin{matrix} 2 & -7 \\ -1 & 4 \\ \end{matrix} \right]$

Therefore,
$A^{-1}=\dfrac{1}{1}.\left[ \begin{matrix} 2 & -7 \\ -1 & 4 \\ \end{matrix} \right]\\
A^{-1}=\left[ \begin{matrix} 2 & -7 \\ -1 & 4 \\ \end{matrix} \right]$

So, option is A correct.

Note:
${{A}^{-1}}$ exists only when $|A|\ne 0$. We have to remember the formula for ${{A}^{-1}}$. Sometimes students make mistakes while solving $adjA$ and $|A|$ for a matrix. If the inverse of a matrix exists, we can find the adjoint of the given matrix and divide it by the determinant of the matrix.