91Ó°ÊÓ

\(\begin{aligned} 2 x_{1}+x_{2}-3 x_{3} &=1 \\ x_{1}-2 x_{2}+x_{3} &=0 \\ 3 x_{1}+4 x_{2}-2 x_{3} &=-5 \end{aligned}\)

Suppose that \(M \in \mathbb{M}_{n \times n}(F)\) 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 \(\operatorname{det}(M)=\operatorname{det}(A)\).

\(x_{1}-x_{2}+4 x_{3}=-4\) \(-8 x_{1}+3 x_{2}+x_{3}=8\) \(2 x_{1}-x_{2}+x_{3}=0\)

Prove that if \(B\) is the matrix obtained by interchanging the rows of a \(2 \times 2\) matrix \(A\), then \(\operatorname{det}(B)=-\operatorname{det}(A)\).

In Exercises 5- 12, evaluate the determinant of the given matrix by cofactor expansion along the indicated row. $$ \left(\begin{array}{rrr} 0 & 1 & 2 \\ -1 & 0 & -3 \\ 2 & 3 & 0 \end{array}\right) $$ along the first row

\(x_{1}-x_{2}+4 x_{3}=-2\) \(-8 x_{1}+3 x_{2}+x_{3}=0\) \(2 x_{1}-x_{2}+x_{3}=6\)

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.g1/pgMdpX for a solution.

Evaluate the determinant of the given matrix by cofactor expansion along the indicated row. $$ \left(\begin{array}{rrr} 1 & 0 & 2 \\ 0 & 1 & 5 \\ -1 & 3 & 0 \end{array}\right) $$ along the first row

Prove that if the two columns of \(A \in \mathrm{M}_{2 \times 2}(F)\) are identical, then \(\operatorname{det}(A)=0\). C$.

Prove that \(\operatorname{det}\left(A^{t}\right)=\operatorname{det}(A)\) for any \(A \in \mathrm{M}_{2 \times 2}(F)\).