Verify whether the coefficient matrix of the following system of equations is diagonalizable or not: 6x - 2y + 2z = 0, -2x + 3y - z = 0, 2x - y + 3z = 0.
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In Problems, determine whether the given matrix $\mathbf{A}$ is diagonalizable. If so, find the matrix $\mathbf{P}$ that diagonalizes $\mathbf{A}$ and the diagonal matrix $\mathbf{D}$ such that $\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}$. $$ \left(\begin{array}{lll} 1 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array}\right) $$
Matrices
Diagonalization
In Problems, determine whether the given matrix $\mathbf{A}$ is diagonalizable. If so, find the matrix $\mathbf{P}$ that diagonalizes $\mathbf{A}$ and the diagonal matrix $\mathbf{D}$ such that $\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}$. $$ \left(\begin{array}{rrrr} 4 & 2 & -1 & 4 \\ 0 & 2 & 0 & 0 \\ 1 & 3 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{array}\right) $$
In Problems, determine whether the given matrix $\mathbf{A}$ is diagonalizable. If so, find the matrix $\mathbf{P}$ that diagonalizes $\mathbf{A}$ and the diagonal matrix $\mathbf{D}$ such that $\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}$. $$ \left(\begin{array}{rrr} 1 & 0 & 1 \\ 0 & -1 & 3 \\ 0 & 0 & 2 \end{array}\right) $$
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