Question

Let A be a 2 × 2 matrix whose eigenvalues are 3 and 4, and associated eigenvectors are [−1 1]
and [4 1], respectively. Without computation, find a diagonal matrix D that is similar to A, and singular matrix P such that P−1AP = D. (Enter each matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list.)

(D, P) = ?

Answer 1

eigenvalues are 3, 4 --> diagonal matrix D is
\[\left[\begin{matrix}3&0\\0&4\end{matrix}\right]\]
just put eigenvalues into the main diagonal line.

Answer 2

P is arranged by the corresponding eigenvectors of 3, 4 respectively. Pay attention, if you put 3 first in D, then eigenvector corresponding to 3, that is [-1,1]. Don't mess up the order(very important)
so, \[P=\left[\begin{matrix}-1&4\\1&1\end{matrix}\right]\]

Answer 3

Thank you so much.........;)