Exercise · ZW-DIAGONALIZATION-007 · ★★☆ Intermediate
With \(A=\begin{pmatrix}4&1\\2&3\end{pmatrix}\), \(A=PDP^{-1}\) and \(D=\operatorname{diag}(5,2)\), what is the diagonal matrix of \(A^5\)?
Choose one option.
A \(\operatorname{diag}(25,10)\) B \(\operatorname{diag}(3125,32)\) C \(\operatorname{diag}(3125,10)\) D \(\operatorname{diag}(625,16)\) Show answer B. \(\operatorname{diag}(3125,32)\)← Back to the question
Explanation \(A^5=PD^5P^{-1}\) with \(D^5=\operatorname{diag}(5^5,2^5)=\operatorname{diag}(3125,32)\).
A square matrix \(A\) is diagonalizable if it is similar to a diagonal matrix \(D\), that is, if it can be written as an invertible matrix \(P\) times \(D\) times the inverse of \(P\).
\[A=PDP^{-1}\]
More practice problems: diagonalization