[Math] Find eigenvalues given A and eigenvectors

eigenvalues-eigenvectorslinear algebramatrices

I have the following problem:enter image description here

I know how to compute the eigenvectors given the matrix and then finding eigenvalues. I could turn A into a triangular matrix and then compute for lambdas, but I wanted to know if there was another procedure by using the eigenvectors and A to find the eigenvalues.

Any help is appreciated.

Thanks!

Best Answer

$$ \begin{bmatrix} 13 & 2 & -18\\ 14 & 1 & -18\\ 10 & 2 & -15 \end{bmatrix}\cdot \begin{bmatrix} -1\\ -1\\ -1 \end{bmatrix}= \begin{bmatrix} 3\\ 3\\ 3 \end{bmatrix}=-3\cdot \begin{bmatrix} -1\\ -1\\ -1 \end{bmatrix}. $$

Thus, $-3$ is an eigenvalue of $A$. Try the same strategy for the other two eigenvectors.

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