Chapter 3

Find the trace of the given matrix. \(\left[\begin{array}{cc}7 & 5 \\ -5 & -4\end{array}\right]\)

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ll}1 & 0 \\ 0 & 9\end{array}\right]\)

Find the determinant of the \(2 \times 2\) matrix. \(\left[\begin{array}{cc}-1 & -7 \\ -5 & 9\end{array}\right]\)

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

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. \(A=\left[\begin{array}{cc}-8 & 16 \\ 10 & -20\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}-48 \\ 60\end{array}\right]\)

Find the determinant of the \(2 \times 2\) matrix. \(\left[\begin{array}{cc}-10 & -1 \\ -4 & 7\end{array}\right]\)

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. \(A=\left[\begin{array}{cc}0 & -6 \\ 9 & -10\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}6 \\ -17\end{array}\right]\)

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{cc}13 & -3 \\ -3 & 1\end{array}\right]\)

Find the trace of the given matrix. \(\left[\begin{array}{cc}-6 & 0 \\ -10 & 9\end{array}\right]\)

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