91Ó°ÊÓ

Evaluate the determinant of the given matrix by any legitimate method. $$ \left(\begin{array}{rrrr} 1 & 0 & -2 & 3 \\ -3 & 1 & 1 & 2 \\ 0 & 4 & -1 & 1 \\ 2 & 3 & 0 & 1 \end{array}\right) $$

\(^{\dagger}\) Prove that if \(M \in M_{n \times n}(F)\) can be written in the form $$ M=\left(\begin{array}{ll} A & B \\ O & C \end{array}\right) $$ where \(A\) and \(C\) are square matrices, then \(\operatorname{det}(M)=\operatorname{det}(A) \cdot \operatorname{det}(C)\). Visit goo.gl/4sG3iv for a solution.

Let \(A \in \mathrm{M}_{n \times n}(F)\) be nonzero. For any \(m(1 \leq m \leq n)\), an \(m \times m\) submatrix is obtained by deleting any \(n-m\) rows and any \(n-m\) columns of \(A\). (a) Let \(k(1 \leq k \leq n)\) denote the largest integer such that some \(k \times k\) submatrix has a nonzero determinant. Prove that \(\operatorname{rank}(A)=k\). (b) Conversely, suppose that \(\operatorname{rank}(A)=k\). Prove that there exists a \(k \times k\) submatrix with a nonzero determinant.

Prove that the determinant of an upper triangular matrix is the product of its diagonal entries.

Find the classical adjoint of each of the following matrices. (a) \(\left(\begin{array}{ll}A_{11} & A_{12} \\ A_{21} & A_{22}\end{array}\right)\) (b) \(\left(\begin{array}{lll}4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4\end{array}\right)\) (c) \(\left(\begin{array}{rrr}-4 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 5\end{array}\right)\) (d) \(\left(\begin{array}{lll}3 & 6 & 7 \\ 0 & 4 & 8 \\ 0 & 0 & 5\end{array}\right)\) (e) \(\left(\begin{array}{ccc}1-i & 0 & 0 \\ 4 & 3 i & 0 \\ 2 i & 1+4 i & -1\end{array}\right)\) (f) \(\left(\begin{array}{rrr}7 & 1 & 4 \\ 6 & -3 & 0 \\ -3 & 5 & -2\end{array}\right)\) (g) \(\left(\begin{array}{rrr}-1 & 2 & 5 \\ 8 & 0 & -3 \\ 4 & 6 & 1\end{array}\right)\) (h) \(\left(\begin{array}{ccc}3 & 2+i & 0 \\ -1+i & 0 & i \\ 0 & 1 & 3-2 i\end{array}\right)\)

Prove that if \(A \in \mathrm{M}_{n \times n}(F)\) has two identical columns, then \(\operatorname{det}(A)=0\).

Prove that if \(E\) is an elementary matrix, then \(\operatorname{det}\left(E^{t}\right)=\operatorname{det}(E) .\) Visit goo.gl/6ZoU5Z for a solution.