Chapter 3

Find the determinant of the given matrix using cofactor expansion along any row or column you choose. \(\left[\begin{array}{ccc}1 & 2 & 3 \\ -5 & 0 & 3 \\ 4 & 0 & 6\end{array}\right]\)

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{cc}-7 & 4 \\ 4 & -6\end{array}\right]\)

Matrices \(A\) and \(\vec{b}\) are given. (a) Give \(\operatorname{det}(A)\) and \(\operatorname{det}\left(A_{i}\right)\) for all \(i\). (b) Use Cramer's Rule to solve \(A \vec{x}=\vec{b}\). If Cramer's Rule cannot be used to find the solution, then state whether or not a solution exists. \(A=\left[\begin{array}{cc}7 & -7 \\ -7 & 9\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}28 \\ -26\end{array}\right]\)

Find the determinant of the \(2 \times 2\) matrix. \(\left[\begin{array}{cc}10 & 7 \\ 8 & 9\end{array}\right]\)

Find the trace of the given matrix. \(\left[\begin{array}{cc}1 & -5 \\ 9 & 5\end{array}\right]\)

Find the trace of the given matrix. \(\left[\begin{array}{cc}-3 & -10 \\ -6 & 4\end{array}\right]\)

Find the determinant of the \(2 \times 2\) matrix. \(\left[\begin{array}{cc}6 & -1 \\ -7 & 8\end{array}\right]\)

Matrices \(A\) and \(\vec{b}\) are given. (a) Give \(\operatorname{det}(A)\) and \(\operatorname{det}\left(A_{i}\right)\) for all \(i\). (b) Use Cramer's Rule to solve \(A \vec{x}=\vec{b}\). If Cramer's Rule cannot be used to find the solution, then state whether or not a solution exists. \(A=\left[\begin{array}{cc}9 & 5 \\ -4 & -7\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}-45 \\ 20\end{array}\right]\)

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{cc}3 & 1 \\ -7 & 8\end{array}\right]\)

Find the determinant of the given matrix using cofactor expansion along any row or column you choose. \(\left[\begin{array}{ccc}-4 & 4 & -4 \\ 0 & 0 & -3 \\ -2 & -2 & -1\end{array}\right]\)