Question

If matrix A is invertible, is `(A^2)^-1 = (A^-1)^2`?

Answer 1

No take for example the matrix

#\[A = \left( {\begin{array}{*{20}{c}} 1&0\\ 1&1 \end{array}} \right)\]#

we can easily find that

#\[{\left( {{A^2}} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} 1&0\\ { - 2}&1 \end{array}} \right)\]#

but

#\[{\left( {{A^{ - 1}}} \right)^2} = \left( {\begin{array}{*{20}{c}} 1&0\\ 1&1 \end{array}} \right)\]#

hence it is not always true that

`\[{\left( {{A^{ - 1}}} \right)^2} = {\left( {{A^2}} \right)^{ - 1}}\]`