Problem 18
Verify Theorem 13 by: 1\. Showing that \(\operatorname{tr}(A)+\operatorname{tr}(B)=\operatorname{tr}(A+B)\) and 2\. Showing that \(\operatorname{tr}(A B)=\operatorname{tr}(B A)\). \(\begin{aligned} A &=\left[\begin{array}{ccc}-8 & -10 & 10 \\ 10 & 5 & -6 \\\ -10 & 1 & 3\end{array}\right] \\ B &=\left[\begin{array}{ccc}-10 & -4 & -3 \\\ -4 & -5 & 4 \\ 3 & 7 & 3\end{array}\right] \end{aligned}\)
Problem 18
Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}4 & 0 & 0 \\ -2 & -7 & 0 \\ 4 & -2 & 5\end{array}\right]\)
Problem 19
Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}-3 & -4 & -5 \\ 0 & -3 & 5 \\ 0 & 0 & -3\end{array}\right]\)
Problem 19
Verify Theorem 13 by: 1\. Showing that \(\operatorname{tr}(A)+\operatorname{tr}(B)=\operatorname{tr}(A+B)\) and 2\. Showing that \(\operatorname{tr}(A B)=\operatorname{tr}(B A)\). \(\begin{aligned} A &=\left[\begin{array}{ccc}-10 & 7 & 5 \\ 7 & 7 & -5 \\ 8 & -9 & 2\end{array}\right] \\ B &=\left[\begin{array}{ccc}-3 & -4 & 9 \\ 4 & -1 & -9 \\ -7 & -8 & 10\end{array}\right] \end{aligned}\)
Problem 19
Find the determinant of the given matrix using cofactor expansion along the first row. 13\. \(\left[\begin{array}{ccc}3 & 2 & 3 \\ -6 & 1 & -10 \\ -8 & -9 & -9\end{array}\right]\) 14\. \(\left[\begin{array}{ccc}8 & -9 & -2 \\ -9 & 9 & -7 \\ 5 & -1 & 9\end{array}\right]\) 15\. \(\left[\begin{array}{ccc}-4 & 3 & -4 \\ -4 & -5 & 3 \\ 3 & -4 & 5\end{array}\right]\) 16\. \(\left[\begin{array}{ccc}1 & -2 & 1 \\ 5 & 5 & 4 \\ 4 & 0 & 0\end{array}\right]\) 17\. \(\left[\begin{array}{ccc}1 & -4 & 1 \\ 0 & 3 & 0 \\ 1 & 2 & 2\end{array}\right]\) 18\. \(\left[\begin{array}{ccc}3 & -1 & 0 \\ -3 & 0 & -4 \\ 0 & -1 & -4\end{array}\right]\) 19\. \(\left[\begin{array}{ccc}-5 & 0 & -4 \\ 2 & 4 & -1 \\ -5 & 0 & -4\end{array}\right]\) 20\. \(\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 1 & 1\end{array}\right]\)
Problem 20
Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{cccc}6 & -7 & 2 & 6 \\ 0 & -8 & -1 & 0 \\ 0 & 0 & 1 & -7\end{array}\right]\)
Problem 20
Find the determinant of the given matrix using cofactor expansion along the first row. \(\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 1 & 1\end{array}\right]\)
Problem 21
Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1\end{array}\right]\)
Problem 21
Find the determinant of the given matrix using cofactor expansion along the first row. \(\left[\begin{array}{cccc}0 & 0 & -1 & -1 \\ 1 & 1 & 0 & 1 \\ 1 & 1 & -1 & 0 \\\ -1 & 0 & 1 & 0\end{array}\right]\)
Problem 22
Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{ccc}6 & -4 & -5 \\ -4 & 0 & 2 \\ -5 & 2 & -2\end{array}\right]\)
