For this question, matrix A is as follows:
$$ \begin{pmatrix} 0 & -1 & 0 \\ 2 & -3 & 0 \\ 0 & 0 & -1 \\ \end{pmatrix} .$$
I have found the eigenvalues of A; however, how do I find matrix B without guessing until I find the correct eigenvalue of $-3$? Is there a more systematic way of doing this?
Multiplying $A$ by $$B=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 3 \\ \end{pmatrix}$$ will switch the first two rows of $A$ so the product is an upper triangular matrix,
and it will multiply the third row by $3,$ so $-3$ is an element on the diagonal and thus an eigenvalue.