## Determine Eigenvalues, Eigenvectors, Diagonalizable From a Partial Information of a Matrix

## Problem 180

Suppose the following information is known about a $3\times 3$ matrix $A$.

\[A\begin{bmatrix}

1 \\

2 \\

1

\end{bmatrix}=6\begin{bmatrix}

1 \\

2 \\

1

\end{bmatrix},

\quad

A\begin{bmatrix}

1 \\

-1 \\

1

\end{bmatrix}=3\begin{bmatrix}

1 \\

-1 \\

1

\end{bmatrix}, \quad

A\begin{bmatrix}

2 \\

-1 \\

0

\end{bmatrix}=3\begin{bmatrix}

1 \\

-1 \\

1

\end{bmatrix}.\]

**(a)** Find the eigenvalues of $A$.

**(b)** Find the corresponding eigenspaces.

**(c)** In each of the following questions, you must give a correct reason (based on the theory of eigenvalues and eigenvectors) to get full credit.

Is $A$ a diagonalizable matrix?

Is $A$ an invertible matrix?

Is $A$ an idempotent matrix?

(*Johns Hopkins University Linear Algebra Exam*)

Read solution