Question

# Find all the characteristic values and vectors of the matrix.$\left( \begin{array}{ll}{3} & {4} \\ {5} & {2}\end{array}\right)$

Solution

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Step 1
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Determine 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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