Eigenvalues and Eigenvectors

Practice Questions

Linear Algebra › Eigenvalues and Eigenvectors

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A matrix has $\begin{Bmatrix} 1+i, 1-i, -1+i, -1-i \end{Bmatrix}$ as its set of eigenvalues.

Give the set of eigenvalues of $A^{-1}$ .

A $4 \times 4$ matrix has $\begin{Bmatrix} 1+i, 1-i, -1+i, -1-i \end{Bmatrix}$ as its set of eigenvalues.

True, false, or indeterminate: the matrix is singular.

is a nonsingular real matrix with four eigenvalues: $\left {x, \frac{1}{x} , -x , - \frac{1}{x} \right }$ .

True or false: $A^{-1}$ must have these same four eigenvalues.

A $4 \times 4$ matrix has $\begin{Bmatrix} 1+i, 1-i, -1+i, -1-i \end{Bmatrix}$ as its set of eigenvalues.

Calculate $\det A$ .

is a nonsingular real matrix with four eigenvalues: $\left { \lambda, -\lambda , \lambda i, -\lambda i \right }$ .

True or false: $A^{-1}$ must have these same four eigenvalues.

is an eigenvalue of a nonsingular real matrix .

True or false: It follows that is an eigenvalue of $A ^{-1}$ .

$\begin{align*}&\text{Find any and all eigenvalues for the matrix }\&A=\begin{bmatrix}6&-13\8&-19\end{bmatrix}\end{align*}$

Suppose we have a square matrix with real-valued entries with only positive eigenvalues. Is invertible? Why or why not?

$A= \begin{bmatrix} 2 & 3 & 6 \ 0 & 4 & 4 \ 2 & 0 & 4 \end{bmatrix}$ .

Is an eigenvalue of , and if so, what is the dimension of its eigenspace?

$A = \begin{bmatrix} a & 4 & -2 \ 1 & 3 & 0 \ - 6 & 4 & a \end{bmatrix}$

Evaluate so that the sum of the eigenvalues of is 10.

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