Problem 1
A matrix \(A\) is given. For each, (a) Find the eigenvalues of \(A,\) and for each eigenvalue, find an eigenvector. (b) Do the same for \(A^{T}\). (c) Do the same for \(A^{-1}\). (d) Find \(\operatorname{tr}(A)\). (e) Find det \((A)\). Use Theorem 19 to verify your results. $$\left[\begin{array}{cc}0 & 4 \\ -1 & 5\end{array}\right]$$
Problem 1
A matrix \(A\) and one of its eigenvectors are given. Find the eigenvalue of \(A\) for the given eigenvector. $$ \begin{array}{l} A=\left[\begin{array}{cc} 9 & 8 \\ -6 & -5 \end{array}\right] \\ \vec{x}=\left[\begin{array}{c} -4 \\ 3 \end{array}\right] \end{array} $$
Problem 2
A matrix \(A\) and one of its eigenvectors are given. Find the eigenvalue of \(A\) for the given eigenvector. $$ \begin{array}{l} A=\left[\begin{array}{cc} 19 & -6 \\ 48 & -15 \end{array}\right] \\ \vec{x}=\left[\begin{array}{l} 1 \\ 3 \end{array}\right] \end{array} $$
Problem 3
A matrix \(A\) and one of its eigenvectors are given. Find the eigenvalue of \(A\) for the given eigenvector. $$ \begin{array}{l} A=\left[\begin{array}{cc} 1 & -2 \\ -2 & 4 \end{array}\right] \\ \vec{x}=\left[\begin{array}{l} 2 \\ 1 \end{array}\right] \end{array} $$
Problem 3
A matrix \(A\) is given. For each, (a) Find the eigenvalues of \(A,\) and for each eigenvalue, find an eigenvector. (b) Do the same for \(A^{T}\). (c) Do the same for \(A^{-1}\). (d) Find \(\operatorname{tr}(A)\). (e) Find det \((A)\). Use Theorem 19 to verify your results. $$\left[\begin{array}{cc}5 & 30 \\ -1 & -6\end{array}\right]$$
Problem 4
A matrix \(A\) is given. For each, (a) Find the eigenvalues of \(A,\) and for each eigenvalue, find an eigenvector. (b) Do the same for \(A^{T}\). (c) Do the same for \(A^{-1}\). (d) Find \(\operatorname{tr}(A)\). (e) Find det \((A)\). Use Theorem 19 to verify your results. $$\left[\begin{array}{ll}-4 & 72 \\ -1 & 13\end{array}\right]$$
Problem 4
A matrix \(A\) and one of its eigenvectors are given. Find the eigenvalue of \(A\) for the given eigenvector. $$ \begin{array}{l} A=\left[\begin{array}{ccc} -11 & -19 & 14 \\ -6 & -8 & 6 \\ -12 & -22 & 15 \end{array}\right] \\ \vec{x}=\left[\begin{array}{l} 3 \\ 2 \\ 4 \end{array}\right] \end{array} $$
Problem 5
A matrix \(A\) and one of its eigenvectors are given. Find the eigenvalue of \(A\) for the given eigenvector. $$ \begin{array}{l} A=\left[\begin{array}{ccc} -7 & 1 & 3 \\ 10 & 2 & -3 \\ -20 & -14 & 1 \end{array}\right] \\ \vec{x}=\left[\begin{array}{c} 1 \\ -2 \\ 4 \end{array}\right] \end{array} $$
Problem 6
A matrix \(A\) is given. For each, (a) Find the eigenvalues of \(A,\) and for each eigenvalue, find an eigenvector. (b) Do the same for \(A^{T}\). (c) Do the same for \(A^{-1}\). (d) Find \(\operatorname{tr}(A)\). (e) Find det \((A)\). Use Theorem 19 to verify your results. $$\left[\begin{array}{ccc}0 & 25 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & -3\end{array}\right]$$
Problem 6
A matrix \(A\) and one of its eigenvectors are given. Find the eigenvalue of \(A\) for the given eigenvector. $$ A=\left[\begin{array}{ccc} -12 & -10 & 0 \\ 15 & 13 & 0 \\ 15 & 18 & -5 \end{array}\right] $$ $$ \vec{x}=\left[\begin{array}{c} -1 \\ 1 \\ 1 \end{array}\right] $$
