91Ó°ÊÓ

Find the eigenvalues and eigenvectors of A geometrically \(A=\left[\begin{array}{rr}-1 & 0 \\ 0 & 1\end{array}\right]\) (reflection in the \(y\) -axis)

Determine whether \(A\) is diagonalizable and, if so, find an invertible matrix \(P\) and a diagonal matrix \(D\) such that \(P^{-1} A P=D\). $$A=\left[\begin{array}{rrr} 1 & 2 & 1 \\ -1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]$$

Find the eigenvalues and eigenvectors of A geometrically \(.A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$$ \text { (reflection in the line) } y=x\)

Compute the determinants in using cofactor expansion along any row or column that seems convenient $$\left|\begin{array}{rrrr} 2 & 0 & 3 & -1 \\ 1 & 0 & 2 & 2 \\ 0 & -1 & 1 & 4 \\ 2 & 0 & 1 & -3 \end{array}\right|$$

Determine whether \(A\) is diagonalizable and, if so, find an invertible matrix \(P\) and a diagonal matrix \(D\) such that \(P^{-1} A P=D\). $$A=\left[\begin{array}{llll} 2 & 0 & 0 & 2 \\ 0 & 3 & 2 & 1 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$

Use the power method to approximate the dominant eigenvalue and eigenvector of \(A .\) Use the given initial vector \(\mathbf{x}_{0},\) the specified number of iterations \(k,\) and three-decimal-place accuracy. $$A=\left[\begin{array}{lll}3 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 1 & 3\end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l}1 \\ 1 \\\ 1\end{array}\right], k=6$$

\(A\) is a \(2 \times 2\) matrix with eigenvectors \(\mathbf{v}_{1}=\left[\begin{array}{r}1 \\ -1\end{array}\right]\) and \(\mathbf{v}_{2}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) corresponding to eigenvalues \(\lambda_{1}=\frac{1}{2}\) and \(\lambda_{2}=2,\) respectively,and \(\mathbf{x}=\left[\begin{array}{l}5 \\ 1\end{array}\right]\) Find \(A^{10} \mathbf{x}\)

Find the eigenvalues and eigenvectors of A geometrically \(A=\left[\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right]\) (projection onto the \(x\) -axis)

Determine whether \(A\) is diagonalizable and, if so, find an invertible matrix \(P\) and a diagonal matrix \(D\) such that \(P^{-1} A P=D\). $$A=\left[\begin{array}{rrrr} 2 & 0 & 0 & 4 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & -2 & 0 \\ 0 & 0 & 0 & -2 \end{array}\right]$$

Compute the determinants in using cofactor expansion along any row or column that seems convenient $$\left|\begin{array}{llll} 0 & 0 & 0 & a \\ 0 & 0 & b & c \\ 0 & d & e & f \\ g & h & i & j \end{array}\right|$$