91Ó°ÊÓ

Find the inverse matrix to each given matrix if the inverse matrix exists. $$ A=\left[\begin{array}{rrr} -1 & 0 & -1 \\ 0 & -2 & 0 \\ -1 & 1 & 2 \end{array}\right] $$

Let $$ A=\left[\begin{array}{rr} -1 & 1 \\ 0 & 2 \end{array}\right] $$ (a) Show that $$ \mathbf{u}_{1}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] \quad \text { and } \quad \mathbf{u}_{2}=\left[\begin{array}{l} 1 \\ 3 \end{array}\right] $$ are eigenvectors of \(A\) and that \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are linearly independent. (b) Represent $$ \mathbf{x}=\left[\begin{array}{r} 1 \\ -3 \end{array}\right] $$ as a linear combination of \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\). (c) Use your results in (a) and (b) to compute \(A^{20} \mathbf{x}\).

Let $$ A=\left[\begin{array}{rr} -1 & -2 \\ -4 & 1 \end{array}\right] $$ (a) Show that $$ \mathbf{u}_{1}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right] \quad \text { and } \quad \mathbf{u}_{2}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right] $$ are eigenvectors of \(A\) and that \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are linearly independent. (b) Represent $$ \mathbf{x}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right] $$ as a linear combination of \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\). (c) Use your results in (a) and (b) to compute \(A^{10} \mathbf{x}\).

Find the inverse matrix to each given matrix if the inverse matrix exists. $$ A=\left[\begin{array}{rrr} -1 & 0 & 2 \\ -1 & -2 & 3 \\ 0 & 2 & -1 \end{array}\right] $$

Let $$ A=\left[\begin{array}{rr} -1 & 0 \\ 2 & 1 \end{array}\right] $$ Find $$ A^{15}\left[\begin{array}{l} 2 \\ 0 \end{array}\right] $$ without using a calculator.

Let $$ A=\left[\begin{array}{rr} 4 & 3 \\ 2 & -1 \end{array}\right] . $$ Find $$ A^{30}\left[\begin{array}{l} -4 \\ -2 \end{array}\right] $$ without using a calculator.

Let $$ A=\left[\begin{array}{rr} 5 & 7 \\ -2 & -4 \end{array}\right] $$ Find $$ A^{20}\left[\begin{array}{l} -3 \\ -2 \end{array}\right] $$ without using a calculator.