91Ó°ÊÓ

Determine all minors and cofactors of the given matrix. $$A=\left[\begin{array}{rr} 1 & -3 \\ 2 & 4 \end{array}\right]$$

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\left|\begin{array}{rrr} 1 & 2 & 3 \\ 2 & 6 & 4 \\ 3 & -5 & 2 \end{array}\right|$$

Determine the number of inversions and the parity of the given permutation. (5,4,3,2,1).

Evaluate the determinant of the given matrix by first using elementary row operations to reduce it to upper triangular form. $$\left|\begin{array}{rrr} 1 & -1 & 2 & 4 \\ 3 & 1 & 2 & 4 \\ -1 & 1 & 3 & 2 \\ 2 & 1 & 4 & 2 \end{array}\right|$$

Use the cofactor expansion theorem to evaluate the given determinant along the specified row or column. \(\left|\begin{array}{rrr}2 & 1 & -4 \\ 7 & 1 & 3 \\ 1 & 5 & -2\end{array}\right|,\) row 2

Use the cofactor expansion theorem to evaluate the given determinant along the specified row or column. \(\left|\begin{array}{rrr}0 & 2 & -3 \\ -2 & 0 & 5 \\ 3 & -5 & 0\end{array}\right|,\) row 3.

Use Cramer's rule to determine the unique solution for \(x\) to the system \(A x=b\) for the given matrix \(A\) and vector \(\mathbf{b}\). $$A=\left[\begin{array}{rl}2 & 8 \\\\-2 & 4\end{array}\right], \mathbf{b}=\left[\begin{array}{r}0 \\\\-3\end{array}\right].$$

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 \(\left(A^{-1} B^{T}\right)^{-1}.\)

Use the adjoint method to determine \(A^{-1}\) for the given matrix \(A.\) $$A=\left[\begin{array}{rrrr} 0 & -3 & 2 & 2 \\ 0 & 1 & 1 & 1 \\ 1 & 2 & 3 & -4 \\ 1 & 0 & 0 & 5 \end{array}\right]$$.

True or False: Given any real number \(r\) and any \(3 \times 3\) matrix \(A\) whose entries are all nonzero, it is always possible to change at most one entry of \(A\) to get a matrix \(B\) with \(\operatorname{det}(B)=r.\)