sympy import symbols, Matrix, det

Define the variable lambda

lam = symbols('lambda')

Define the matrix A - λI

matrix = Matrix([ [2 - lam, 2, 1], [1, 3 - lam, 1], [1, 2, 2 - lam] ])

Calculate the determinant

determinant = det(matrix) determinantThe determinant of the matrix $A - \lambda I$ is:

$\text{det}(A - \lambda I) = -\lambda^3 + 7\lambda^2 - 11\lambda + 5$

Detailed Steps:

1. Matrix Setup: The matrix $A - \lambda I$ is: 2 - \lambda & 2 & 1 \\ 1 & 3 - \lambda & 1 \\ 1 & 2 & 2 - \lambda \end{bmatrix}$$
2. Determinant Formula: The determinant of a $3 \times 3$ matrix is calculated as: $\text{det} = a(ei − fh) − b(di − fg) + c(dh − eg)$ where the matrix is: a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$
3. Apply the Formula: Substituting the elements: - 2\big((1)(2-\lambda) - (1)(1)\big) + 1\big((1)(2) - (1)(3-\lambda)\big)$$
4. Simplify: Expanding and simplifying results in: $-\lambda^3 + 7\lambda^2 - 11\lambda + 5$

Let me know if you'd like me to expand the determinant manually or clarify further!

Follow-up Questions:

1. What are the eigenvalues of this matrix?
2. Can you calculate the characteristic polynomial from this determinant?
3. How does the determinant relate to the invertibility of the matrix $A - \lambda I$?
4. What happens when $\lambda$ equals one of the roots of the determinant?
5. How is the determinant used to compute eigenvectors?

Tip:

When solving determinants, always expand along the row or column with the most zeros to simplify calculations!