For [3 1 00 3 00 0 3] find algebraic and geometric multiplicities of eigen values

To find the algebraic and geometric multiplicities of the eigenvalues for the matrix \(\begin{bmatrix} 3 & 1 & 0 \\ 3 & 0 & 0 \\ 0 & 3 & 0 \end{bmatrix}\), we need to first compute its eigenvalues and then determine their multiplicities.

### Step 1: Find the Eigenvalues

To find the eigenvalues, we need to solve the characteristic equation:

\[
\det(A - \lambda I) = 0
\]

where \(A = \begin{bmatrix} 3 & 1 & 0 \\ 3 & 0 & 0 \\ 0 & 3 & 0 \end{bmatrix}\) and \(I\) is the identity matrix.

First, compute \(A - \lambda I\):

\[
A - \lambda I = \begin{bmatrix} 3 - \lambda & 1 & 0 \\ 3 & -\lambda & 0 \\ 0 & 3 & -\lambda \end{bmatrix}
\]

Now, find the determinant of \(A - \lambda I\):

\[
\det(A - \lambda I) = \begin{vmatrix} 3 - \lambda & 1 & 0 \\ 3 & -\lambda & 0 \\ 0 & 3 & -\lambda \end{vmatrix}
\]

Since the third column only has one non-zero entry, expand along this column:

\[
\det(A - \lambda I) = (-1)^{3+3}(-\lambda) \begin{vmatrix} 3 - \lambda & 1 \\ 3 & -\lambda \end{vmatrix}
\]

Simplify the determinant:

\[
\det(A - \lambda I) = -\lambda \left((3 - \lambda)(-\lambda) - (1)(3)\right)
\]

\[
= -\lambda \left(\lambda^2 - 3\lambda - 3\right)
\]

Thus, the characteristic polynomial is:

\[
\lambda^3 - 3\lambda^2 - 3\lambda = 0
\]

Factor the polynomial:

\[
\lambda(\lambda^2 - 3\lambda - 3) = 0
\]

So, the eigenvalues are found by solving:

1. \(\lambda = 0\)
2. \(\lambda^2 - 3\lambda - 3 = 0\)

Solve the quadratic equation \(\lambda^2 - 3\lambda - 3 = 0\) using the quadratic formula \(\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

\[
\lambda = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-3)}}{2(1)} = \frac{3 \pm \sqrt{9 + 12}}{2}
\]

\[
\lambda = \frac{3 \pm \sqrt{21}}{2}
\]

So, the eigenvalues are:

\[
\lambda_1 = 0, \quad \lambda_2 = \frac{3 + \sqrt{21}}{2}, \quad \lambda_3 = \frac{3 - \sqrt{21}}{2}
\]

### Step 2: Determine Algebraic and Geometric Multiplicities

- **

Explore the algebraic and geometric multiplicities of eigenvalues for the matrix [3 1 00 3 00 0 3]. Learn how to compute eigenvalues using characteristic polynomials and determine their multiplicities. Suitable for advanced college-level mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation