## G3 - Eigenvalues

#### Example 1 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -4 & 1 \\ 7 & 2 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -4 & 1 \\ 7 & 2 \end{array}\right]$$ is $$\lambda^{2} + 2 \lambda - 15$$.

The eigenvalues of $$\left[\begin{array}{cc} -4 & 1 \\ 7 & 2 \end{array}\right]$$ are $$-5$$ and $$3$$.

#### Example 2 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 9 & 2 \\ -28 & -9 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 9 & 2 \\ -28 & -9 \end{array}\right]$$ is $$\lambda^{2} - 25$$.

The eigenvalues of $$\left[\begin{array}{cc} 9 & 2 \\ -28 & -9 \end{array}\right]$$ are $$5$$ and $$-5$$.

#### Example 3 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 7 & 2 \\ -18 & -6 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 7 & 2 \\ -18 & -6 \end{array}\right]$$ is $$\lambda^{2} - \lambda - 6$$.

The eigenvalues of $$\left[\begin{array}{cc} 7 & 2 \\ -18 & -6 \end{array}\right]$$ are $$3$$ and $$-2$$.

#### Example 4 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 1 & 2 \\ 0 & -5 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 1 & 2 \\ 0 & -5 \end{array}\right]$$ is $$\lambda^{2} + 4 \lambda - 5$$.

The eigenvalues of $$\left[\begin{array}{cc} 1 & 2 \\ 0 & -5 \end{array}\right]$$ are $$-5$$ and $$1$$.

#### Example 5 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 6 & 1 \\ -20 & -6 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 6 & 1 \\ -20 & -6 \end{array}\right]$$ is $$\lambda^{2} - 16$$.

The eigenvalues of $$\left[\begin{array}{cc} 6 & 1 \\ -20 & -6 \end{array}\right]$$ are $$4$$ and $$-4$$.

#### Example 6 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -1 & 1 \\ 2 & 0 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -1 & 1 \\ 2 & 0 \end{array}\right]$$ is $$\lambda^{2} + \lambda - 2$$.

The eigenvalues of $$\left[\begin{array}{cc} -1 & 1 \\ 2 & 0 \end{array}\right]$$ are $$-2$$ and $$1$$.

#### Example 7 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 7 & 1 \\ -11 & -5 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 7 & 1 \\ -11 & -5 \end{array}\right]$$ is $$\lambda^{2} - 2 \lambda - 24$$.

The eigenvalues of $$\left[\begin{array}{cc} 7 & 1 \\ -11 & -5 \end{array}\right]$$ are $$6$$ and $$-4$$.

#### Example 8 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 0 & 2 \\ 8 & 0 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 0 & 2 \\ 8 & 0 \end{array}\right]$$ is $$\lambda^{2} - 16$$.

The eigenvalues of $$\left[\begin{array}{cc} 0 & 2 \\ 8 & 0 \end{array}\right]$$ are $$-4$$ and $$4$$.

#### Example 9 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 4 & 1 \\ -18 & -5 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 4 & 1 \\ -18 & -5 \end{array}\right]$$ is $$\lambda^{2} + \lambda - 2$$.

The eigenvalues of $$\left[\begin{array}{cc} 4 & 1 \\ -18 & -5 \end{array}\right]$$ are $$1$$ and $$-2$$.

#### Example 10 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -1 & 1 \\ 21 & 3 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -1 & 1 \\ 21 & 3 \end{array}\right]$$ is $$\lambda^{2} - 2 \lambda - 24$$.

The eigenvalues of $$\left[\begin{array}{cc} -1 & 1 \\ 21 & 3 \end{array}\right]$$ are $$-4$$ and $$6$$.

#### Example 11 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 0 & 2 \\ 4 & 2 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 0 & 2 \\ 4 & 2 \end{array}\right]$$ is $$\lambda^{2} - 2 \lambda - 8$$.

The eigenvalues of $$\left[\begin{array}{cc} 0 & 2 \\ 4 & 2 \end{array}\right]$$ are $$-2$$ and $$4$$.

#### Example 12 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 4 & 1 \\ 2 & 5 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 4 & 1 \\ 2 & 5 \end{array}\right]$$ is $$\lambda^{2} - 9 \lambda + 18$$.

The eigenvalues of $$\left[\begin{array}{cc} 4 & 1 \\ 2 & 5 \end{array}\right]$$ are $$3$$ and $$6$$.

#### Example 13 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -2 & 1 \\ 4 & 1 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -2 & 1 \\ 4 & 1 \end{array}\right]$$ is $$\lambda^{2} + \lambda - 6$$.

The eigenvalues of $$\left[\begin{array}{cc} -2 & 1 \\ 4 & 1 \end{array}\right]$$ are $$-3$$ and $$2$$.

#### Example 14 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 1 & 2 \\ 12 & -1 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 1 & 2 \\ 12 & -1 \end{array}\right]$$ is $$\lambda^{2} - 25$$.

The eigenvalues of $$\left[\begin{array}{cc} 1 & 2 \\ 12 & -1 \end{array}\right]$$ are $$-5$$ and $$5$$.

#### Example 15 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -1 & 1 \\ 5 & 3 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -1 & 1 \\ 5 & 3 \end{array}\right]$$ is $$\lambda^{2} - 2 \lambda - 8$$.

The eigenvalues of $$\left[\begin{array}{cc} -1 & 1 \\ 5 & 3 \end{array}\right]$$ are $$-2$$ and $$4$$.

#### Example 16 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -1 & 2 \\ 3 & 0 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -1 & 2 \\ 3 & 0 \end{array}\right]$$ is $$\lambda^{2} + \lambda - 6$$.

The eigenvalues of $$\left[\begin{array}{cc} -1 & 2 \\ 3 & 0 \end{array}\right]$$ are $$-3$$ and $$2$$.

#### Example 17 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 6 & 2 \\ -12 & -8 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 6 & 2 \\ -12 & -8 \end{array}\right]$$ is $$\lambda^{2} + 2 \lambda - 24$$.

The eigenvalues of $$\left[\begin{array}{cc} 6 & 2 \\ -12 & -8 \end{array}\right]$$ are $$4$$ and $$-6$$.

#### Example 18 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 3 & 2 \\ 9 & 0 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 3 & 2 \\ 9 & 0 \end{array}\right]$$ is $$\lambda^{2} - 3 \lambda - 18$$.

The eigenvalues of $$\left[\begin{array}{cc} 3 & 2 \\ 9 & 0 \end{array}\right]$$ are $$-3$$ and $$6$$.

#### Example 19 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 0 & 1 \\ 12 & 1 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 0 & 1 \\ 12 & 1 \end{array}\right]$$ is $$\lambda^{2} - \lambda - 12$$.

The eigenvalues of $$\left[\begin{array}{cc} 0 & 1 \\ 12 & 1 \end{array}\right]$$ are $$-3$$ and $$4$$.

#### Example 20 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 2 & 2 \\ 4 & 0 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 2 & 2 \\ 4 & 0 \end{array}\right]$$ is $$\lambda^{2} - 2 \lambda - 8$$.

The eigenvalues of $$\left[\begin{array}{cc} 2 & 2 \\ 4 & 0 \end{array}\right]$$ are $$-2$$ and $$4$$.

#### Example 21 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 7 & 2 \\ -6 & 0 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 7 & 2 \\ -6 & 0 \end{array}\right]$$ is $$\lambda^{2} - 7 \lambda + 12$$.

The eigenvalues of $$\left[\begin{array}{cc} 7 & 2 \\ -6 & 0 \end{array}\right]$$ are $$3$$ and $$4$$.

#### Example 22 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 0 & 1 \\ 2 & -1 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 0 & 1 \\ 2 & -1 \end{array}\right]$$ is $$\lambda^{2} + \lambda - 2$$.

The eigenvalues of $$\left[\begin{array}{cc} 0 & 1 \\ 2 & -1 \end{array}\right]$$ are $$-2$$ and $$1$$.

#### Example 23 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 9 & 2 \\ -42 & -11 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 9 & 2 \\ -42 & -11 \end{array}\right]$$ is $$\lambda^{2} + 2 \lambda - 15$$.

The eigenvalues of $$\left[\begin{array}{cc} 9 & 2 \\ -42 & -11 \end{array}\right]$$ are $$3$$ and $$-5$$.

#### Example 24 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 4 & 2 \\ 0 & 2 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 4 & 2 \\ 0 & 2 \end{array}\right]$$ is $$\lambda^{2} - 6 \lambda + 8$$.

The eigenvalues of $$\left[\begin{array}{cc} 4 & 2 \\ 0 & 2 \end{array}\right]$$ are $$2$$ and $$4$$.

#### Example 25 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -4 & 1 \\ 2 & -5 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -4 & 1 \\ 2 & -5 \end{array}\right]$$ is $$\lambda^{2} + 9 \lambda + 18$$.

The eigenvalues of $$\left[\begin{array}{cc} -4 & 1 \\ 2 & -5 \end{array}\right]$$ are $$-6$$ and $$-3$$.

#### Example 26 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -3 & 1 \\ 12 & 1 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -3 & 1 \\ 12 & 1 \end{array}\right]$$ is $$\lambda^{2} + 2 \lambda - 15$$.

The eigenvalues of $$\left[\begin{array}{cc} -3 & 1 \\ 12 & 1 \end{array}\right]$$ are $$-5$$ and $$3$$.

#### Example 27 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 1 & 1 \\ 9 & 1 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 1 & 1 \\ 9 & 1 \end{array}\right]$$ is $$\lambda^{2} - 2 \lambda - 8$$.

The eigenvalues of $$\left[\begin{array}{cc} 1 & 1 \\ 9 & 1 \end{array}\right]$$ are $$-2$$ and $$4$$.

#### Example 28 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -3 & 1 \\ -3 & -7 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -3 & 1 \\ -3 & -7 \end{array}\right]$$ is $$\lambda^{2} + 10 \lambda + 24$$.

The eigenvalues of $$\left[\begin{array}{cc} -3 & 1 \\ -3 & -7 \end{array}\right]$$ are $$-4$$ and $$-6$$.

#### Example 29 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 9 & 1 \\ -42 & -8 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 9 & 1 \\ -42 & -8 \end{array}\right]$$ is $$\lambda^{2} - \lambda - 30$$.

The eigenvalues of $$\left[\begin{array}{cc} 9 & 1 \\ -42 & -8 \end{array}\right]$$ are $$6$$ and $$-5$$.

#### Example 30 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -4 & 1 \\ 2 & -5 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -4 & 1 \\ 2 & -5 \end{array}\right]$$ is $$\lambda^{2} + 9 \lambda + 18$$.

The eigenvalues of $$\left[\begin{array}{cc} -4 & 1 \\ 2 & -5 \end{array}\right]$$ are $$-6$$ and $$-3$$.

#### Example 31 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 0 & 2 \\ -4 & -6 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 0 & 2 \\ -4 & -6 \end{array}\right]$$ is $$\lambda^{2} + 6 \lambda + 8$$.

The eigenvalues of $$\left[\begin{array}{cc} 0 & 2 \\ -4 & -6 \end{array}\right]$$ are $$-4$$ and $$-2$$.

#### Example 32 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -2 & 1 \\ 0 & -3 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -2 & 1 \\ 0 & -3 \end{array}\right]$$ is $$\lambda^{2} + 5 \lambda + 6$$.

The eigenvalues of $$\left[\begin{array}{cc} -2 & 1 \\ 0 & -3 \end{array}\right]$$ are $$-3$$ and $$-2$$.

#### Example 33 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 8 & 1 \\ -6 & 3 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 8 & 1 \\ -6 & 3 \end{array}\right]$$ is $$\lambda^{2} - 11 \lambda + 30$$.

The eigenvalues of $$\left[\begin{array}{cc} 8 & 1 \\ -6 & 3 \end{array}\right]$$ are $$5$$ and $$6$$.

#### Example 34 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 5 & 1 \\ -7 & -3 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 5 & 1 \\ -7 & -3 \end{array}\right]$$ is $$\lambda^{2} - 2 \lambda - 8$$.

The eigenvalues of $$\left[\begin{array}{cc} 5 & 1 \\ -7 & -3 \end{array}\right]$$ are $$4$$ and $$-2$$.

#### Example 35 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 6 & 1 \\ -6 & 1 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 6 & 1 \\ -6 & 1 \end{array}\right]$$ is $$\lambda^{2} - 7 \lambda + 12$$.

The eigenvalues of $$\left[\begin{array}{cc} 6 & 1 \\ -6 & 1 \end{array}\right]$$ are $$4$$ and $$3$$.

#### Example 36 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 6 & 1 \\ -11 & -6 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 6 & 1 \\ -11 & -6 \end{array}\right]$$ is $$\lambda^{2} - 25$$.

The eigenvalues of $$\left[\begin{array}{cc} 6 & 1 \\ -11 & -6 \end{array}\right]$$ are $$5$$ and $$-5$$.

#### Example 37 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 7 & 2 \\ -5 & 0 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 7 & 2 \\ -5 & 0 \end{array}\right]$$ is $$\lambda^{2} - 7 \lambda + 10$$.

The eigenvalues of $$\left[\begin{array}{cc} 7 & 2 \\ -5 & 0 \end{array}\right]$$ are $$5$$ and $$2$$.

#### Example 38 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 3 & 1 \\ 3 & 5 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 3 & 1 \\ 3 & 5 \end{array}\right]$$ is $$\lambda^{2} - 8 \lambda + 12$$.

The eigenvalues of $$\left[\begin{array}{cc} 3 & 1 \\ 3 & 5 \end{array}\right]$$ are $$2$$ and $$6$$.

#### Example 39 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 1 & 2 \\ 8 & 1 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 1 & 2 \\ 8 & 1 \end{array}\right]$$ is $$\lambda^{2} - 2 \lambda - 15$$.

The eigenvalues of $$\left[\begin{array}{cc} 1 & 2 \\ 8 & 1 \end{array}\right]$$ are $$-3$$ and $$5$$.

#### Example 40 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 0 & 1 \\ -15 & -8 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 0 & 1 \\ -15 & -8 \end{array}\right]$$ is $$\lambda^{2} + 8 \lambda + 15$$.

The eigenvalues of $$\left[\begin{array}{cc} 0 & 1 \\ -15 & -8 \end{array}\right]$$ are $$-3$$ and $$-5$$.

#### Example 41 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -2 & 1 \\ 0 & -3 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -2 & 1 \\ 0 & -3 \end{array}\right]$$ is $$\lambda^{2} + 5 \lambda + 6$$.

The eigenvalues of $$\left[\begin{array}{cc} -2 & 1 \\ 0 & -3 \end{array}\right]$$ are $$-3$$ and $$-2$$.

#### Example 42 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -4 & 1 \\ 14 & 1 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -4 & 1 \\ 14 & 1 \end{array}\right]$$ is $$\lambda^{2} + 3 \lambda - 18$$.

The eigenvalues of $$\left[\begin{array}{cc} -4 & 1 \\ 14 & 1 \end{array}\right]$$ are $$-6$$ and $$3$$.

#### Example 43 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 9 & 2 \\ -15 & -2 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 9 & 2 \\ -15 & -2 \end{array}\right]$$ is $$\lambda^{2} - 7 \lambda + 12$$.

The eigenvalues of $$\left[\begin{array}{cc} 9 & 2 \\ -15 & -2 \end{array}\right]$$ are $$3$$ and $$4$$.

#### Example 44 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 5 & 1 \\ -21 & -5 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 5 & 1 \\ -21 & -5 \end{array}\right]$$ is $$\lambda^{2} - 4$$.

The eigenvalues of $$\left[\begin{array}{cc} 5 & 1 \\ -21 & -5 \end{array}\right]$$ are $$2$$ and $$-2$$.

#### Example 45 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -1 & 2 \\ 6 & -2 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -1 & 2 \\ 6 & -2 \end{array}\right]$$ is $$\lambda^{2} + 3 \lambda - 10$$.

The eigenvalues of $$\left[\begin{array}{cc} -1 & 2 \\ 6 & -2 \end{array}\right]$$ are $$-5$$ and $$2$$.

#### Example 46 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 0 & 1 \\ 10 & 3 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 0 & 1 \\ 10 & 3 \end{array}\right]$$ is $$\lambda^{2} - 3 \lambda - 10$$.

The eigenvalues of $$\left[\begin{array}{cc} 0 & 1 \\ 10 & 3 \end{array}\right]$$ are $$-2$$ and $$5$$.

#### Example 47 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} -3 & 2 \\ 7 & 2 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} -3 & 2 \\ 7 & 2 \end{array}\right]$$ is $$\lambda^{2} + \lambda - 20$$.

The eigenvalues of $$\left[\begin{array}{cc} -3 & 2 \\ 7 & 2 \end{array}\right]$$ are $$-5$$ and $$4$$.

#### Example 48 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 3 & 2 \\ 6 & -1 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 3 & 2 \\ 6 & -1 \end{array}\right]$$ is $$\lambda^{2} - 2 \lambda - 15$$.

The eigenvalues of $$\left[\begin{array}{cc} 3 & 2 \\ 6 & -1 \end{array}\right]$$ are $$-3$$ and $$5$$.

#### Example 49 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 9 & 2 \\ -36 & -9 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 9 & 2 \\ -36 & -9 \end{array}\right]$$ is $$\lambda^{2} - 9$$.

The eigenvalues of $$\left[\begin{array}{cc} 9 & 2 \\ -36 & -9 \end{array}\right]$$ are $$3$$ and $$-3$$.

#### Example 50 π

Explain how to find the eigenvalues of the matrix $$\left[\begin{array}{cc} 4 & 1 \\ 6 & 3 \end{array}\right]$$.

The characteristic polynomial of $$\left[\begin{array}{cc} 4 & 1 \\ 6 & 3 \end{array}\right]$$ is $$\lambda^{2} - 7 \lambda + 6$$.
The eigenvalues of $$\left[\begin{array}{cc} 4 & 1 \\ 6 & 3 \end{array}\right]$$ are $$1$$ and $$6$$.