Question
For the following matrices, check whether there exists an invertible matrix P such that P-1AP is diagonal. When such a P exists, find P.
i)
Answer :
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Skip in 10
i) (A = \begin{bmatrix}0 & 1 & -3 \ 2 & -1 & 6 \ 1 & -1 & 4\end{bmatrix}) Step 1: Find the characteristic polynomial (\det(A - \lambda I) = 0). [ A - \lambda I = \begin{bmatrix}-\lambda & 1 & -3 \ 2 & -1-\lambda & 6 \ 1 & -1 & 4-\lambda\end{bmatrix} ] [ \det(A - \lambda I) = -\lambda\begin{vmatrix}-1-\lambda & 6 \ -1 & 4-\lambda\end{vmatrix} - 1 \begin{vmatrix}2 & 6 \ 1 & 4-\lambda\end{vmatrix} + (-3)\begin{vmatrix}2 & -1-\lambda \ 1 & -1\end{vmatrix} ] Compute each minor: 1. (\begin{vmatrix}-1-\lambda & 6 \ -1 & 4-\lambda\end{vmatrix} = (-1-\lambda)(4-\lambda) - (-1)(6) = (-4 + \lambda - 4\lambda + \lambda^2) + 6 = \lambda^2 - 3\lambda + 2) 2. (\begin{vmatrix}2 & 6 \ 1 & 4-\lambda\end{vmatrix} = 2(4-\lambda) - 6(1) = 8 - 2\lambda - 6 = 2 - 2\lambda) 3. (\begin{vmatrix}2 & -1-\lambda \ 1 & -1\end{vmatrix} = 2(-1) - 1(-1-\lambda) = -2 + 1 + \lambda = \lambda - 1) Combine with cofactors: [ -\lambda (\lambda^2 - 3\lambda + 2) - 1(2 - 2\lambda) - 3(\lambda - 1) = -\lambda^3 + 3\lambda^2 - 2\lambda - 2 + 2\lambda - 3\lambda + 3 ] Simplify: [ -\lambda^3 + 3\lambda^2 - 3\lambda +
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Related Questions
For the following matrices, check whether there exists an invertible matrix P such that P-1AP is diagonal. When such a P exists, find P.
i)
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