91Ó°ÊÓ

Compute the determinants in Exercises \(7-15\) using cofactor expansion along any row or column that seems convenient. $$\left|\begin{array}{rrr}\cos \theta & \sin \theta & \tan \theta \\\0 & \cos \theta & -\sin \theta \\\0 & \sin \theta & \cos \theta \end{array}\right|$$

Compute the determinants in Exercises \(7-15\) using cofactor expansion along any row or column that seems convenient. $$\left|\begin{array}{lll}a & b & 0 \\\0 & a & b \\\a & 0 & b\end{array}\right|$$

Calculate the positive eigenvalue and \(a\) corresponding positive eigenvector of the Leslie matrix \(L\). $$L=\left[\begin{array}{ll} 0 & 2 \\ 0.5 & 0 \end{array}\right]$$

Show that \(\lambda\) is an eigenvalue of \(A\) and find one eigenvector corresponding to this eigenvalue. $$A=\left[\begin{array}{rrr} 1 & 0 & 2 \\ -1 & 1 & 1 \\ 2 & 0 & 1 \end{array}\right], \lambda=-1$$

Compute (a) the characteristic polynomial of \(A,(b)\) the eigenvalues of \(A,(c)\) a basis for each eigenspace of \(A,\) and (d) the algebraic and geometric multiplicity of each eigenvalue. $$A=\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 3 & 0 \\ -2 & 1 & 2 & -1 \end{array}\right]$$

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. $$A=\left[\begin{array}{ll} 7 & 2 \\ 2 & 3 \end{array}\right], \mathbf{x}_{0}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right], k=6$$

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

Calculate the positive eigenvalue and \(a\) corresponding positive eigenvector of the Leslie matrix \(L\). $$L=\left[\begin{array}{ll} 1 & 1.5 \\ 0.5 & 0 \end{array}\right]$$

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

Compute (a) the characteristic polynomial of \(A,(b)\) the eigenvalues of \(A,(c)\) a basis for each eigenspace of \(A,\) and (d) the algebraic and geometric multiplicity of each eigenvalue. $$A=\left[\begin{array}{llll} 4 & 0 & 1 & 0 \\ 0 & 4 & 1 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 3 & 0 \end{array}\right]$$