91Ó°ÊÓ

Use Cramer's rule to solve the given linear system. $$\begin{aligned} 3 x_{1}+x_{2}+2 x_{3} &=-1, \\ 2 x_{1}-x_{2}+x_{3} &=-1, \\ 5 x_{2}+5 x_{3} &=-5. \end{aligned}$$

Let \(A\) and \(B\) be \(4 \times 4\) matrices such that \(\operatorname{det}(A)=5\) and \(\operatorname{det}(B)=3 .\) Compute the determinant of the given matrix. $$B^{-1} A^{-1}$$

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrrrr}0 & 0 & 0 & 8 & 4 \\ 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 2 & 0 & 0 \\ 2 & -3 & 0 & 0 & 0 \\ 4 & -2 & 0 & 0 & 0\end{array}\right]\).

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C},(\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{rrr} 2 & -3 & 0 \\ 2 & 1 & 5 \\ 0 & -1 & 2 \end{array}\right]$$

Evaluate the determinant of the given matrix function. \(A(t)=\left[\begin{array}{cc}e^{6 t} & e^{4 t} \\ 6 e^{6 t} & 4 e^{4 t}\end{array}\right]\).

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C},(\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{rrr} -2 & 3 & -1 \\ 2 & 1 & 5 \\ 0 & 2 & 3 \end{array}\right]$$

Let \(A\) and \(B\) be \(4 \times 4\) matrices such that \(\operatorname{det}(A)=5\) and \(\operatorname{det}(B)=3 .\) Compute the determinant of the given matrix. $$B^{-1}(2 A) B^{T}$$

Let \(A\) and \(B\) be \(4 \times 4\) matrices such that \(\operatorname{det}(A)=5\) and \(\operatorname{det}(B)=3 .\) Compute the determinant of the given matrix. $$(4 B)^{3}$$

Evaluate the determinant of the given matrix function. \(A(t)=\left[\begin{array}{lll}\sin t & \cos t & 1 \\ \cos t & -\sin t & 0 \\\ \sin t & -\cos t & 0\end{array}\right]\).

Find (a) \(\operatorname{det}(A),\) (b) the matrix of cofactors \(M_{C},(\mathrm{c})\) adj \((A),\) and, if possible, \((\mathrm{d}) A^{-1}.\) $$A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 3 & -1 & 4 \\ 5 & 1 & 7 \end{array}\right]$$