Question

Determine the eigenvalues of the following matrix... (in comments)

Answer 1

\[A=\left[\begin{matrix}-10 & 5 &-9&0 \\ 0 & -10&10&1\\0&0&-10&2\\0&0&0&-5\end{matrix}\right]\] So to find the eigenvalues, I know that I would need to find the determinant of \[\left[\begin{matrix}-10-\lambda & 5&-9&0 \\ 0 & -10-\lambda&10&1\\0&0&-10-\lambda&2\\0&0&0&-5- \lambda\end{matrix}\right]\] ....And this is where I come unstuck. How do I find the determinant of a 4x4 matrix??

Answer 2

The eigenvalues are -10 and -5. You can read them off from the diagonal in this case, because the matrix is upper triangular.

Answer 3

And you can find the determinant of a 4x4 matrix by using Laplace's formula. Its pretty easy in this case, but as I said you don't need to do it here.