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Q.
If $A$ is an invertible matrix, then $\left(A^{-1}\right)^{\prime}$ is equal to
Determinants
Solution:
Since, $A$ is an invertible matrix, so it is non-singular.
We know that, $|A|=\left|A^{\prime}\right|$. But $|A| \neq 0$. So, $\left|A^{\prime}\right| \neq 0$ i.e., $A^{\prime}$ is invertible matrix.
Also, we know that, $A A^{-1}=A^{-1} A=1$.
Taking transpose on both sides, we get
$\left(A^{-1}\right) A^{\prime}=A^{\prime}\left(A^{-1}\right)^{\prime}=(I)^{\prime}=I$
Hence, $\left(A^{-1}\right)$ is inverse of $A^{\prime}$, i.e., $\left(A^{\prime}\right)^{-1}=\left(A^{-1}\right)$.