## Related questions with answers

Question

Find all the characteristic values and vectors of the matrix.

$\left( \begin{array}{ll}{3} & {4} \\ {5} & {2}\end{array}\right)$

Solution

VerifiedAnswered 2 years ago

Answered 2 years ago

Step 1

1 of 4Determine the eigenvalues of the matrix $\mathbf{A}$.

$\begin{align*} \det\left(\mathbf{A}-\lambda\mathbf{I}\right) &= 0\\ \begin{vmatrix} 3-\lambda & 4\\ 5 & 2-\lambda \end{vmatrix} &= 0\\ \left(3-\lambda\right)\left(2-\lambda\right)-20&=0\\ 6-3\lambda-2\lambda+\lambda^2-20&=0\\ \lambda^2-5\lambda-14&=0\\ \left(\lambda-7\right)\left(\lambda+2\right)&=0\\\\ \lambda_1=7\;\;\text{or}\;\;\lambda_2&=-2 \end{align*}$

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