Diagonalization and Matrix Powers

00:00 6 marks

Explain the process of computing \(A^n\) for a diagonalizable 3×3 matrix A using the diagonalization formula \(A=PDP^{-1}\). Subsequently, apply this method to compute \(A^5\) for the matrix \(A=\begin{pmatrix} 2 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}\).

Hint Hint

Hint

Recall that if \(A=PDP^{-1}\), then \(A^n=PD^nP^{-1}\), and that computing powers of a diagonal matrix is straightforward.

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\(A^5=\begin{pmatrix} 32 & 211 & 0 \\ 0 & 243 & 0 \\ 0 & 0 & 1024 \end{pmatrix}\)

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Diagonalization and Matrix Powers

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Diagonalization and Matrix Powers

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