91Ó°ÊÓ

Use elementary row operations together with the Cofactor Expansion Theorem to evaluate the given determinant. $$\left|\begin{array}{rrrr} -2 & 0 & 1 & 1 \\ 1 & 2 & 2 & 0 \\ -4 & 4 & 6 & 1 \\ -1 & 1 & 0 & 5 \end{array}\right|$$

Let $$A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 2 & 1 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 1 \\ 5 & -2 \\ 4 & 7 \end{array}\right], \quad C=\left[\begin{array}{rrr} 1 & 0 & 5 \\ 3 & -1 & 4 \\ 2 & -2 & 6 \end{array}\right]$$. Compute the determinants, where possible. $$\operatorname{det}\left(B C C^{T}\right)$$

If \(A=\left[\begin{array}{rrr}1 & -1 & 2 \\ 3 & 1 & 4 \\ 0 & 1 & 3\end{array}\right],\) find \(\operatorname{det}(A),\) and use properties of determinants to find \(\operatorname{det}\left(A^{-1}\right)\) and \(\operatorname{det}(-3 A)\)

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrr}5 & -3 & 0 \\ 1 & 4 & -1 \\ -8 & 2 & -2\end{array}\right]\).

Assume that \(A\) and \(B\) be \(3 \times 3\) matrices with \(\operatorname{det}(A)=3\) and \(\operatorname{det}(B)=-4 .\) Compute the specified determinant. \(\operatorname{det}\left(B^{5}\right)\)

Assume that \(A\) and \(B\) be \(3 \times 3\) matrices with \(\operatorname{det}(A)=3\) and \(\operatorname{det}(B)=-4 .\) Compute the specified determinant. \(\operatorname{det}\left(B^{-1} A B\right)^{2}\)

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrr}-2 & -4 & 1 \\ 6 & 1 & 1 \\ -2 & -1 & 3\end{array}\right]\).

Let \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right],\) and let \(B=\left[\begin{array}{ll}5 & 4 \\ 1 & 1\end{array}\right] .\) Use the adjoint method to find \(B^{-1}\) and then determine \(\left(A^{-1} B^{T}\right)^{-1}.\)

Evaluate the determinant of the given matrix. \(A=\left[\begin{array}{rrr}0 & 0 & -3 \\ 0 & 4 & 3 \\ -2 & 1 & 5\end{array}\right]\).

Assume that \(A\) and \(B\) be \(3 \times 3\) matrices with \(\operatorname{det}(A)=3\) and \(\operatorname{det}(B)=-4 .\) Compute the specified determinant. \(\operatorname{det}(C),\) where \(C\) is obtained from matrix \(B\) by interchanging the last two columns and multiplying the first column by 4