Question

Let A = [[4, 0, 1], [2, 3, 2], [1, 0, 4]]. You may take for granted that the eigenvalues of A are 5 and 3, and that a basis for the eigenspace of A corresponding to the eigenvalue 5 is {[[1], [2], [1]]}. Determine whether or not A is diagonalizable. If not, explain why not; if so, explicitly find an invertible matrix P and a diagonal matrix D such that A = PDP^-1 (do not compute P^-1).

          Let
A = [[4, 0, 1], [2, 3, 2], [1, 0, 4]].
You may take for granted that the eigenvalues of A are 5 and 3, and that a basis for the eigenspace of A corresponding to the eigenvalue 5 is
{[[1], [2], [1]]}.
Determine whether or not A is diagonalizable. If not, explain why not; if so, explicitly find an invertible matrix P and a diagonal matrix D such that A = PDP^-1 (do not compute P^-1).

Let
A = [[4, 0, 1], [2, 3, 2], [1, 0, 4]].
You may take for granted that the eigenvalues of A are 5 and 3, and that a basis for the eigenspace of A corresponding to the eigenvalue 5 is
[[1], [2], [1]].
Determine whether or not A is diagonalizable. If not, explain why not; if so, explicitly find an invertible matrix P and a diagonal matrix D such that A = PDP^-1 (do not compute P^-1).

Added by Joshua E.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

07/03/2023

Step 1

Let's find the eigenvectors corresponding to these eigenvalues. For the eigenvalue 5, we have: $$ (A - 5I)v = 0 \Rightarrow \begin{pmatrix} -3 & 2 & 1 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}v = 0 $$ We can see that the eigenspace corresponding to the Show more…

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Let A = [[4, 0, 1], [2, 3, 2], [1, 0, 4]]. You may take for granted that the eigenvalues of A are 5 and 3, and that a basis for the eigenspace of A corresponding to the eigenvalue 5 is {[[1], [2], [1]]}. Determine whether or not A is diagonalizable. If not, explain why not; if so, explicitly find an invertible matrix P and a diagonal matrix D such that A = PDP^-1 (do not compute P^-1).