Diagonalize $$$\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right]$$$

Diagonalize $$$\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right]$$$.

First, find the eigenvalues and eigenvectors (for steps, see eigenvalues and eigenvectors calculator).

Eigenvalue: $$$1$$$, eigenvector: $$$\left[\begin{array}{c}5\\1\end{array}\right]$$$.

Eigenvalue: $$$-2$$$, eigenvector: $$$\left[\begin{array}{c}2\\1\end{array}\right]$$$.

Form the matrix $$$P$$$, whose column $$$i$$$ is eigenvector no. $$$i$$$: $$$P = \left[\begin{array}{cc}5 & 2\\1 & 1\end{array}\right]$$$.

Form the diagonal matrix $$$D$$$ whose element at row $$$i$$$, column $$$i$$$ is eigenvalue no. $$$i$$$: $$$D = \left[\begin{array}{cc}1 & 0\\0 & -2\end{array}\right]$$$.

The matrices $$$P$$$ and $$$D$$$ are such that the initial matrix $$$\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right] = P D P^{-1}$$$.

$$$P^{-1} = \left[\begin{array}{cc}\frac{1}{3} & - \frac{2}{3}\\- \frac{1}{3} & \frac{5}{3}\end{array}\right]$$$ (for steps, see inverse matrix calculator).

$$$P = \left[\begin{array}{cc}5 & 2\\1 & 1\end{array}\right]$$$A

$$$D = \left[\begin{array}{cc}1 & 0\\0 & -2\end{array}\right]$$$A

$$$P^{-1} = \left[\begin{array}{cc}\frac{1}{3} & - \frac{2}{3}\\- \frac{1}{3} & \frac{5}{3}\end{array}\right]\approx \left[\begin{array}{cc}0.333333333333333 & -0.666666666666667\\-0.333333333333333 & 1.666666666666667\end{array}\right]$$$A