Answer
a. False.
b. False.
c. False.
d. False.
Work Step by Step
a. Let $B$ be the result of performing three row interchanges on $A$. Then by Theorem 3(c), we get that $\det B=(-1)^{3}\det A=-\det A$. Hence, the determinant is not the same: its sign is reversed.
b. According to Theorem 2, this is in general true only if the matrix is upper or lower triangular.
c. Theorem 4 and the Invertible Matrix Theorem tell us that the converse is true, i.e., that any equal rows/columns or zero rows/columns in a square matrix will yield a determinant equal to zero. However, there are other reasons a determinant can have a zero determinant; in particular, one row (or column) may be a constant multiple of another row (or column).
d. A trivial counterexample is the $2\times 2$ identity matrix $I_{2}$. This matrix is its own inverse, so we would need $\begin{vmatrix}1&0\\0&1\end{vmatrix}=1=-1$, a blatant contradiction.
