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# Determine whether matrix$\textbf{A}=\begin{bmatrix}4&2&3\\2&1&2\\-1&2&0\end{bmatrix}$is diagonalizable. If it is, determine a matrix $\textbf{P}$ that diagonalizes it and compute $\textbf{P}^{-1}\textbf{AP}$. You can obtain $\textbf{P}^{-1}\textbf{AP}$ directly from careful construction of a diagonal matrix with eigenvalues along the diagonal in the proper order.

Solution

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The eigenvalues are the roots of the characteristic polynomial:

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