91Ó°ÊÓ

Evaluate the determinant of the given matrix by any legitimate method. $$ \left(\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right) $$

Prove that \(\delta: \mathrm{M}_{2 \times 2}(F) \rightarrow F\) is a 2-linear function if and only if it has the form $$ \delta(A)=A_{11} A_{22} a+A_{11} A_{21} b+A_{12} A_{22} c+A_{12} A_{21} d $$ for some scalars \(a, b, c, d \in F\).

\(^{\dagger}\) Prove that if \(A, B \in \mathrm{M}_{n \times n}(F)\) are similar, then \(\operatorname{det}(A)=\operatorname{det}(B)\).

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 \(B=A^{-1}\) ).

Evaluate the determinant of the given matrix by any legitimate method. $$ \left(\begin{array}{rrr} -1 & 3 & 2 \\ 4 & -8 & 1 \\ 2 & 2 & 5 \end{array}\right) $$

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 \(A \in \mathrm{M}_{n \times n}(F)\).

Let \(A, B \in \mathrm{M}_{n \times n}(F)\) be such that \(A B=-B A\). Prove that if \(n\) is odd and \(F\) is not a field of characteristic two, then \(A\) or \(B\) is not invertible.

Evaluate the determinant of the given matrix by any legitimate method. $$ \left(\begin{array}{rrr} 0 & 1 & 1 \\ 1 & 2 & -5 \\ 6 & -4 & 3 \end{array}\right) $$

Prove that the set of all \(n\)-linear functions over a field \(F\) is a vector space over \(F\) under the operations of function addition and scalar multiplication as defined in Example 3 of Section \(1.2\) (p. 9 ).

Evaluate the determinant of the given matrix by any legitimate method. $$ \left(\begin{array}{rrr} 1 & -2 & 3 \\ -1 & 2 & -5 \\ 3 & -1 & 2 \end{array}\right) $$