Matrix with distinct eigenvectors corresponding to eigenvalues.

Question

Preface: This is part of a university assignment. I honestly have no idea what this question is asking me to do other than create a 3x3 matrix of sorts.

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Let $\displaystyle A=\left(\begin{matrix}1&-2 & 0 \\1&4 & 0 \\ 0 & 1 & 0\end{matrix}\right)$

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$\displaystyle \lambda_1 = 3,\hspace{5mm} \lambda_2 = 2,\hspace{5mm} \lambda_3 = 0$

$\displaystyle v_1 = (-3, 3, 1),\hspace{5mm} v_2 = (-4, 2, 1),\hspace{5mm} v_3 = (0, 0, 1)$

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Question: Construct a matrix $P$ whose columns are three distinct eigenvectors corresponding to each distinct eigenvalue you found in the previous question. Find $P^{-1}$, and then verify that $P^{-1}AP$ is equal to a diagonal matrix $D$ having the eigenvalues of $A$ on its diagonal.

Answers in $\LaTeX$ are preferred

Answer 1

Check that $Av_i = \lambda_i v_i$ (most likely you have done so earlier).

What the question is asking to do is let $P= \begin{bmatrix}v_1 & v_2 & v_3 \end{bmatrix}$.

After which compute $P^{-1}$ explicitly, possibly by computing RREF of $[\begin{array}{c|c} P & I\end{array}].$

Compute $P^{-1}AP$ explicitly by multiplying the matrices and see if it is equal to $diag({3,2,0})$