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.