91Ó°ÊÓ

Find \((\mathrm{a})|A|,(\mathrm{b})|B|,(\mathrm{c}) A B,\) and \((\mathrm{d})|A B| .\) Then verify that \(|A||B|=|A B|\) $$A=\left[\begin{array}{rr} -2 & 1 \\ 4 & -2 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 1 \\ 0 & -1 \end{array}\right]$$

Verify that \(\lambda_{i}\) is an eigenvalue of \(A\) and that \(\mathbf{x}_{i}\) is a corresponding eigenvector. $$\begin{array}{l} A=\left[\begin{array}{ll} 1 & 2 \\ 0 & -3 \end{array}\right] ; \quad \lambda_{1}=1, \quad \mathbf{x}_{1}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] \\ \lambda_{2}=-3, \quad \mathbf{x}_{2}=\left[\begin{array}{r} -1 \\ 2 \end{array}\right] \end{array}$$

Which property of determinants is illustrated by the equation? $$\left|\begin{array}{cc} 2 & -6 \\ 1 & -3 \end{array}\right|=0$$

Find the adjoint of the matrix \(A .\) Then use the adjoint to find the inverse of \(A,\) if possible. $$A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]$$

Find the adjoint of the matrix \(A .\) Then use the adjoint to find the inverse of \(A,\) if possible. $$A=\left[\begin{array}{rr} -1 & 0 \\ 0 & 4 \end{array}\right]$$

Verify that \(\lambda_{i}\) is an eigenvalue of \(A\) and that \(\mathbf{x}_{i}\) is a corresponding eigenvector. $$\begin{array}{l} A=\left[\begin{array}{ll} 4 & 3 \\ 1 & 2 \end{array}\right] ; \quad \lambda_{1}=5, \quad \mathbf{x}_{1}=\left[\begin{array}{l} 3 \\ 1 \end{array}\right] \\ \lambda_{2}=1, \quad \mathbf{x}_{2}=\left[\begin{array}{r} -1 \\ 1 \end{array}\right] \end{array}$$

Which property of determinants is illustrated by the equation? $$\left|\begin{array}{rr} -4 & 5 \\ 12 & -15 \end{array}\right|=0$$

Find \((\mathrm{a})|A|,(\mathrm{b})|B|,(\mathrm{c}) A B,\) and \((\mathrm{d})|A B| .\) Then verify that \(|A||B|=|A B|\) $$A=\left[\begin{array}{ll} 1 & 2 \\ 2 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} -1 & 2 \\ 3 & 0 \end{array}\right]$$

Which property of determinants is illustrated by the equation? $$\left|\begin{array}{rrr} 1 & 4 & 2 \\ 0 & 0 & 0 \\ 5 & 6 & -7 \end{array}\right|=0$$

Verify that \(\lambda_{i}\) is an eigenvalue of \(A\) and that \(\mathbf{x}_{i}\) is a corresponding eigenvector. $$\begin{array}{l} A=\left[\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \end{array}\right] ; \quad \lambda_{1}=2, \quad \mathbf{x}_{1}=\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right] \\ \lambda_{2}=0, \quad \mathbf{x}_{2}=\left[\begin{array}{r} -1 \\ 0 \\ 1 \end{array}\right] ; \quad \lambda_{3}=1, \quad \mathbf{x}_{3}=\left[\begin{array}{r} -1 \\ 1 \\ -1 \end{array}\right] \end{array}$$