
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
162.9k+ views
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.
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.
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