Math Snap

STEP 1

1. $A$ is a $3 \times 3$ matrix.
2. We need to find eigenvalues by solving the characteristic equation.
3. Eigenvectors are found by solving $(A - \lambda I)\mathbf{v} = \mathbf{0}$ for each eigenvalue $\lambda$.

STEP 2

1. Find the characteristic polynomial of matrix $A$.
2. Solve the characteristic polynomial to find the eigenvalues.
3. For each eigenvalue, find the corresponding eigenvectors.

STEP 3

To find the characteristic polynomial, we need to compute the determinant of $A - \lambda I$, where $I$ is the identity matrix of the same size as $A$.
The matrix $A - \lambda I$ is:
$$A - \lambda I = \begin{bmatrix} 4 - \lambda & 20 & -20 \\ -6 & -24 - \lambda & 24 \\ -4 & -16 & 16 - \lambda \end{bmatrix}$$ The characteristic polynomial is given by:
$$\det(A - \lambda I) = 0$$

STEP 4

Calculate the determinant of the matrix $A - \lambda I$:
$$\det(A - \lambda I) = \begin{vmatrix} 4 - \lambda & 20 & -20 \\ -6 & -24 - \lambda & 24 \\ -4 & -16 & 16 - \lambda \end{vmatrix}$$ Expanding along the first row:
$$= (4 - \lambda) \begin{vmatrix} -24 - \lambda & 24 \\ -16 & 16 - \lambda \end{vmatrix}
- 20 \begin{vmatrix} -6 & 24 \\ -4 & 16 - \lambda \end{vmatrix}
- 20 \begin{vmatrix} -6 & -24 - \lambda \\ -4 & -16 \end{vmatrix}$$ Calculate each of these 2x2 determinants and simplify to find the characteristic polynomial.

STEP 5

Solve the characteristic polynomial for $\lambda$ to find the eigenvalues.

For each eigenvalue $\lambda$, solve the system $(A - \lambda I)\mathbf{v} = \mathbf{0}$ to find the corresponding eigenvectors.
The solution will provide the distinct eigenvalues and their corresponding basic eigenvectors.