Eigenspace Classifications - GRE Subject Test: Math

Card 0 of 2

Question

Find the eigenvalues of the following matrix, if possible.

$A=\begin{bmatrix} 1&2 \ -2&1 \end{bmatrix}$

Answer

In order to find the eigenvalues of a matrix, apply the following formula:

$det(A-\lambda I) =0$

is the identity matrix.

$\begin{bmatrix} 1&2 \ -2&1 \end{bmatrix}-\lambda \begin{bmatrix} 1&0 \ 0&1 \end{bmatrix}=\begin{bmatrix} 1-\lambda & 2\ -2&1-\lambda \end{bmatrix}$

Compute the determinant and set it equal to zero.

$(1-\lambda)(1-\lambda)-(2)(-2)=0$

$1-2\lambda+\lambda^2+4=0$

$\lambda^2-2\lambda+5=0$

Solve for lambda by using the quadratic formula.

$\lambda_{1,2}=1\pm2i$

Compare your answer with the correct one above

Question

Find the eigenvalues of the following matrix, if possible.

$A=\begin{bmatrix} 1&2 \ -2&1 \end{bmatrix}$

Answer

In order to find the eigenvalues of a matrix, apply the following formula:

$det(A-\lambda I) =0$

is the identity matrix.

$\begin{bmatrix} 1&2 \ -2&1 \end{bmatrix}-\lambda \begin{bmatrix} 1&0 \ 0&1 \end{bmatrix}=\begin{bmatrix} 1-\lambda & 2\ -2&1-\lambda \end{bmatrix}$

Compute the determinant and set it equal to zero.

$(1-\lambda)(1-\lambda)-(2)(-2)=0$

$1-2\lambda+\lambda^2+4=0$

$\lambda^2-2\lambda+5=0$

Solve for lambda by using the quadratic formula.

$\lambda_{1,2}=1\pm2i$

Compare your answer with the correct one above

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