91Ó°ÊÓ

Show that \(A\) is non defective and use Theorem 7.4 .3 to find \(e^{A t}\). $$A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 3 \end{array}\right]$$

Determine the multiplicity of each eigenvalue and a basis for each eigenspace of the given matrix \(A\). Hence, determine the dimension of each eigenspace and state whether the matrix is defective or nondefective. $$A=\left[\begin{array}{rrr} 3 & 0 & 0 \\ 2 & 0 & -4 \\ 1 & 4 & 0 \end{array}\right]$$

Find the Jordan canonical form of each matrix. $$A=\left[\begin{array}{rrr} 5 & 8 & 16 \\ 4 & 1 & 8 \\ -4 & -4 & -11 \end{array}\right].$$ [Hint: The eigenvalues of \(A\) are \(\lambda=1\) and \(\lambda=-3.1]\)

Determine the multiplicity of each eigenvalue and a basis for each eigenspace of the given matrix \(A\). Hence, determine the dimension of each eigenspace and state whether the matrix is defective or nondefective. $$A=\left[\begin{array}{rrr} 7 & -8 & 6 \\ 8 & -9 & 6 \\ 0 & 0 & -1 \end{array}\right]$$

Determine whether the given matrix \(A\) is diagonalizable. Where possible, find a matrix \(S\) such that $$S^{-1} A S=\operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}\right).$$ $$A=\left[\begin{array}{rrr}0 & 2 & -1 \\\\-2 & 0 & -2 \\\1 & 2 & 0\end{array}\right]$$

Find the Jordan canonical form of each matrix. $$A=\left[\begin{array}{rrr} 2 & 1 & 1 \\ 2 & 1 & -2 \\ -1 & 0 & -2 \end{array}\right].$$ [Hint: The eigenvalues of \(A \text { are } \lambda=-1 \text { and } \lambda=3 .]\)

Determine a set of principal axes for the given quadratic form, and reduce the quadratic form to a sum of squares. $$\mathbf{x}^{T} A \mathbf{x}, \quad A=\left[\begin{array}{ll} 5 & 2 \\ 2 & 5 \end{array}\right]$$

Determine the multiplicity of each eigenvalue and a basis for each eigenspace of the given matrix \(A\). Hence, determine the dimension of each eigenspace and state whether the matrix is defective or nondefective. $$A=\left[\begin{array}{rrr} 0 & -1 & -1 \\ -1 & 0 & -1 \\ -1 & -1 & 0 \end{array}\right]$$

Determine whether the given matrix is defective or nondefective. $$A=\left[\begin{array}{rr} 6 & 5 \\ -5 & -4 \end{array}\right]$$

Determine whether the given matrix is defective or nondefective. $$A=\left[\begin{array}{rr} 1 & -2 \\ 5 & 3 \end{array}\right]$$