Answer
(a) the statement is true
(b) the statement is false
(c) the statement is false
Work Step by Step
(a) the statement is true
since $A$ is a nonsingular matrix , then it has the inverse $A^{-1}$ and then
$(AA^{-1})^{-1}=I$ implies $(A^{-1})^{-1}A^{-1}=I$
By multiplying with $A$, we have that
$(A^{-1})^{-1}A^{-1}A=IA=A$
Thus $(A^{-1})^{-1}I=(A^{-1})^{-1}=A$
Therefore $(A^{-1})^{-1}=A$
(b) the statement is false because the matrix $ \left[\begin {array}{cc}
a&b\\
c&d
\end{array}\right]$ is invertible if $ad-bc\ne0$
(c) the statement is false because if $A$ is a singular matrix then the linear system has an infinite number of solutions
