Question Find the eigenvalues of the $2 \times 2$ matrix $A = \left[\begin{array}{cc}3 & 0 \\ -1 & 8\end{array}\right]$ .

Studdy Solution

STEP 1

Assumptions
1. Matrix $A$ is given by $A=\left[\begin{array}{cc}3 & 0 \\ -1 & 8\end{array}\right]$ .
2. We need to find the eigenvalues of matrix $A$ .

STEP 2

The eigenvalues of a matrix $A$ are found by solving the characteristic equation $\det(A - \lambda I) = 0$ , where $I$ is the identity matrix of the same size as $A$ and $\lambda$ represents the eigenvalues.

STEP 3

Write down the identity matrix $I$ of the same size as matrix $A$ .
$I = \left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$

STEP 4

Subtract $\lambda I$ from matrix $A$ to get the matrix $(A - \lambda I)$ .
$(A - \lambda I) = \left[\begin{array}{cc}3 & 0 \\ -1 & 8\end{array}\right] - \lambda \left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right] = \left[\begin{array}{cc}3-\lambda & 0 \\ -1 & 8-\lambda\end{array}\right]$

STEP 5

Find the determinant of the matrix $(A - \lambda I)$ .
$\det(A - \lambda I) = \det\left(\left[\begin{array}{cc}3-\lambda & 0 \\ -1 & 8-\lambda\end{array}\right]\right)$

STEP 6

Calculate the determinant using the formula for a 2x2 matrix: $\det\left(\left[\begin{array}{cc}a & b \\ c & d\end{array}\right]\right) = ad - bc$ .
$\det(A - \lambda I) = (3-\lambda)(8-\lambda) - (0)(-1)$

STEP 7

Expand the determinant expression.
$\det(A - \lambda I) = (3-\lambda)(8-\lambda) = 24 - 3\lambda - 8\lambda + \lambda^2$

STEP 8

Combine like terms in the determinant expression.
$\det(A - \lambda I) = \lambda^2 - 11\lambda + 24$

STEP 9

Now, solve the characteristic equation $\det(A - \lambda I) = 0$ .
$\lambda^2 - 11\lambda + 24 = 0$

STEP 10

Factor the quadratic equation to find the values of $\lambda$ .
$(\lambda - 3)(\lambda - 8) = 0$

STEP 11

Set each factor equal to zero and solve for $\lambda$ .
$\lambda - 3 = 0 \quad \text{or} \quad \lambda - 8 = 0$

STEP 12

Solve for $\lambda$ in each equation.
$\lambda = 3 \quad \text{or} \quad \lambda = 8$
The eigenvalues of matrix $A$ are $\lambda = 3$ and $\lambda = 8$ .
Therefore, the correct answer is a. 3,8.

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