Eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right]$$$

Find the eigenvalues and eigenvectors of $$$\left[\begin{array}{cc}3 & -10\\1 & -4\end{array}\right]$$$.

Start from forming a new matrix by subtracting $$$\lambda$$$ from the diagonal entries of the given matrix: $$$\left[\begin{array}{cc}3 - \lambda & -10\\1 & - \lambda - 4\end{array}\right]$$$.

The determinant of the obtained matrix is $$$\left(\lambda - 1\right) \left(\lambda + 2\right)$$$ (for steps, see determinant calculator).

Solve the equation $$$\left(\lambda - 1\right) \left(\lambda + 2\right) = 0$$$.

The roots are $$$\lambda_{1} = 1$$$, $$$\lambda_{2} = -2$$$ (for steps, see equation solver).

These are the eigenvalues.

Next, find the eigenvectors.

• $$$\lambda = 1$$$

  $$$\left[\begin{array}{cc}3 - \lambda & -10\\1 & - \lambda - 4\end{array}\right] = \left[\begin{array}{cc}2 & -10\\1 & -5\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}5\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  This is the eigenvector.

• $$$\lambda = -2$$$

  $$$\left[\begin{array}{cc}3 - \lambda & -10\\1 & - \lambda - 4\end{array}\right] = \left[\begin{array}{cc}5 & -10\\1 & -2\end{array}\right]$$$

  The null space of this matrix is $$$\left\{\left[\begin{array}{c}2\\1\end{array}\right]\right\}$$$ (for steps, see null space calculator).

  This is the eigenvector.

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

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