Chapter 3

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}0 & 3 & -2 \\ 3 & -4 & 1 \\ -2 & 1 & 0\end{array}\right]\)

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

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}2 & -5 & -3 \\ 5 & 5 & -6 \\ 7 & -4 & -10\end{array}\right]\)

Find the determinant of the given matrix using cofactor expansion along the first row. \(\left[\begin{array}{ccc}-4 & 3 & -4 \\ -4 & -5 & 3 \\ 3 & -4 & 5\end{array}\right]\)

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}0 & -6 & 1 \\ 6 & 0 & 4 \\ -1 & -4 & 0\end{array}\right]\)

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

Verify Theorem 13 by: 1\. Showing that \(\operatorname{tr}(A)+\operatorname{tr}(B)=\operatorname{tr}(A+B)\) and 2\. Showing that \(\operatorname{tr}(A B)=\operatorname{tr}(B A)\). \(A=\left[\begin{array}{cc}0 & -8 \\ 1 & 8\end{array}\right], \quad B=\left[\begin{array}{cc}-4 & 5 \\ -4 & 2\end{array}\right]\)

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

Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}4 & 2 & -9 \\ 5 & -4 & -10 \\ -6 & 6 & 9\end{array}\right]\)

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