Question:
If A is an invertible matrix, then which of the following is not true
(a) $\left(A^{2}\right)^{-1}=\left(A^{-1}\right)^{2}$
(b) $\left|A^{-1}\right|=|A|^{-1}$
(c) $\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$
(d) $|A| \neq 0$
Solution:
(a) $\left(A^{2}\right)^{-1}=\left(A^{-1}\right)^{2}$
We have, $\left|A^{-1}\right|=|A|^{-1},\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}$ and $|A| \neq 0$ all are the properties of the inverse of a matrix $A$
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