Please help with the following linear algebra problem.
Suppose the following information is known about a matrix A: $$ A\begin{bmatrix} 1\\ 2\\ 4 \end{bmatrix} = 9\begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} $$ $$ A\begin{bmatrix} 1\\ -3\\ 9 \end{bmatrix} = 2\begin{bmatrix} 1\\ -3\\ 9 \end{bmatrix} $$ $$ A\begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} = 3\begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} $$
a) Please find eigenvalues.
b) Please find eigenvectors.
c) Is A invertible? Please explain...
d) Is A diagonalizable? Please explain...
Respectfully,
Jorge Maldonado
You can read off two eigenvalues and eigenvectors from the second and third equations. The first equation is presenting you an $x$ which is independent of the two eigenvectors and which is mapped into one of the eigenspaces. This means that there is an eigenvector whose eigenvalue is zero (since we've mapped three independent vectors into the span of only two independent vectors). You can identify this eigenvector by appropriately combining equations 1 and 3. This should get you to your answers for c and d.