Problem 9
Amatrix \(A\) is given. (a) Construct the submatrices used to compute the minors \(A_{1,1}, A_{1,2}\) and \(A_{1.3}\). (b) Find the cofactors \(C_{1,1}, C_{1,2},\) and \(C_{1,3}\) \(.\left[\begin{array}{ccc}-7 & -3 & 10 \\ 3 & 7 & 6 \\ 1 & 6 & 10\end{array}\right]\)
Problem 9
Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{llll}-7 & -8 & 2 & -3\end{array}\right]\)
Problem 9
Find the determinant of the given matrix using cofactor expansion along any row or column you choose. \(\left[\begin{array}{cccc}5 & -5 & 0 & 1 \\ 2 & 4 & -1 & -1 \\ 5 & 0 & 0 & 4 \\\ -1 & -2 & 0 & 5\end{array}\right]\)
Problem 9
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. \(A=\left[\begin{array}{ccc}4 & -4 & 0 \\ 5 & 1 & -1 \\ 3 & -1 & 2\end{array}\right], \quad \vec{b}=\left[\begin{array}{c}16 \\ 22 \\\ 8\end{array}\right]\)
Problem 10
Amatrix \(A\) is given. (a) Construct the submatrices used to compute the minors \(A_{1,1}, A_{1,2}\) and \(A_{1.3}\). (b) Find the cofactors \(C_{1,1}, C_{1,2},\) and \(C_{1,3}\) \(\left[\begin{array}{ccc}-2 & -9 & 6 \\ -10 & -6 & 8 \\ 0 & -3 & -2\end{array}\right]\)
Problem 10
Find the determinant of the given matrix using cofactor expansion along any row or column you choose. \(\left[\begin{array}{cccc}-1 & 3 & 3 & 4 \\ 0 & 0 & 0 & 0 \\ 4 & -5 & -2 & 0 \\\ 0 & 0 & 2 & 0\end{array}\right]\)
Problem 10
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. \(A=\left[\begin{array}{ccc}1 & 0 & -10 \\ 4 & -3 & -10 \\ -9 & 6 & -2\end{array}\right]\) \(\vec{b}=\left[\begin{array}{c}-40 \\ -94 \\ 132\end{array}\right]\)
Problem 10
Find \(A^{T} ;\) make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric. \(\left[\begin{array}{llll}-9 & 8 & 2 & -7\end{array}\right]\)
Problem 11
Amatrix \(A\) is given. (a) Construct the submatrices used to compute the minors \(A_{1,1}, A_{1,2}\) and \(A_{1.3}\). (b) Find the cofactors \(C_{1,1}, C_{1,2},\) and \(C_{1,3}\) \(\left[\begin{array}{ccc}-5 & -3 & 3 \\ -3 & 3 & 10 \\ -9 & 3 & 9\end{array}\right]\)
Problem 11
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. \(A=\left[\begin{array}{ccc}7 & -4 & 25 \\ -2 & 1 & -7 \\ 9 & -7 & 34\end{array}\right]\) \(\vec{b}=\left[\begin{array}{c}-1 \\ -3 \\ 5\end{array}\right]\)
