91Ó°ÊÓ

Find the value of \(k\) that satisfies the following equation: $$ \operatorname{det}\left(\begin{array}{ccc} 3 a_{1} & 3 a_{2} & 3 a_{3} \\ 3 b_{1} & 3 b_{2} & 3 b_{3} \\ 3 c_{1} & 3 c_{2} & 3 c_{3} \end{array}\right)=k \text { det }\left(\begin{array}{ccc} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}\right) \text {. } $$

Find the value of \(k\) that satisfies the following equation: $$ \operatorname{det}\left(\begin{array}{lll} b_{1}+c_{1} & b_{2}+c_{2} & b_{3}+c_{3} \\ a_{1}+c_{1} & a_{2}+c_{2} & a_{3}+c_{3} \\ a_{1}+b_{1} & a_{2}+b_{2} & a_{3}+b_{3} \end{array}\right)=k \operatorname{det}\left(\begin{array}{ccc} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}\right). $$

Prove that an upper triangular \(n \times n\) matrix is invertible if and only if all its diagonal entries are nonzero.

The classical adjoint of a \(2 \times 2\) matrix \(A \in \mathrm{M}_{2 \times 2}(F)\) is the matrix $$ C=\left(\begin{array}{rr} A_{22} & -A_{12} \\ -A_{21} & A_{11} \end{array}\right) . $$ Prove that (a) \(C A=A C=[\operatorname{det}(A)] I\). (b) \(\operatorname{det}(C)=\operatorname{det}(A)\). (c) The classical adjoint of \(A^{t}\) is \(C^{t}\). (d) If \(A\) is invertible, then $A^{-1}=[\operatorname{det}(A)]^{-1}

A matrix \(M \in M_{n \times n}(F)\) is called nilpotent if, for some positive integer \(k, M^{k}=O\), where \(O\) is the \(n \times n\) zero matrix. Prove that if \(M\) is nilpotent, then \(\operatorname{det}(M)=0 .\)

A matrix \(M \in M_{n \times n}(C)\) is called skew-symmetric if \(M^{t}=-M\). Prove that if \(M\) is skew-symmetric and \(n\) is odd, then \(M\) is not invertible. What happens if \(n\) is even?

For \(M \in M_{n \times n}(C)\), let \(\bar{M}\) be the matrix such that \((\bar{M})_{i j}=\overline{M_{i j}}\) for all \(i, j\), where \(\overline{M_{i j}}\) is the complex conjugate of \(M_{i j}\). (a) Prove that \(\operatorname{det}(\bar{M})=\overline{\operatorname{det}(M)}\). (b) A matrix \(Q \in \mathrm{M}_{n \times n}(C)\) is called unitary if \(Q Q^{*}=I\), where \(Q^{*}=\overline{Q^{t}}\). Prove that if \(Q\) is a unitary matrix, then \(|\operatorname{det}(Q)|=1\).

Prove that if \(\delta: M_{n \times n}(F) \rightarrow F\) is an alternating \(n\)-linear function, then there exists a scalar \(k\) such that \(\delta(A)=k \operatorname{det}(A)\) for all \(A \in \mathrm{M}_{n \times n}(F)\).

Use determinants to prove that if \(A, B \in \mathrm{M}_{n \times n}(F)\) are such that \(A B=\) \(I\), then \(A\) is invertible (and hence \(B=A^{-1}\) ).

Suppose that \(M \in M_{n \times n}(F)\) can be written in the form $$ M=\left(\begin{array}{ll} A & B \\ O & I \end{array}\right) $$ where \(A\) is a square matrix. Prove that \(\operatorname{det}(M)=\operatorname{det}(A)\).