Question

For each of the following, answer TRUE or FALSE. If the statement is false in even a single instance, then the answer is FALSE. There is no need to justify your answers to this problem - but you should know either a proof or a counterexample.

a) If \(A\) is an invertible \(4 \times 4\) matrix, then \(\left(A^{T}\right)^{-1}=\left(A^{-1}\right)^{T}\), where \(A^{T}\) denotes the transpose of \(A\).
b) If \(A\) and \(B\) are \(3 \times 3\) matrices, with \(\operatorname{rank}(A)=\operatorname{rank}(B)=2\), then \(\operatorname{rank}(A B)=2\).
c) If \(A\) and \(B\) are invertible \(3 \times 3\) matrices, then \(A+B\) is invertible.
d) If \(A\) is an \(n \times n\) matrix with rank less than \(n\), then for any vector \(b\) the equation \(A x=b\) has an infinite number of solutions.
e) ) If \(A\) is an invertible \(3 \times 3\) matrix and \(\lambda\) is an eigenvalue of \(A\), then \(1 / \lambda\) is an eigenvalue of \(A^{-1}\),