Diagonalize the matrix 
2 1 -1
1 1 -2
1-2 1
Find A?

To diagonalize a matrix, we need to find a matrix \( P \) and a diagonal matrix \( D \) such that:

\[
A = PDP^{-1}
\]

Given matrix \( A \) as:

\[
A = \begin{pmatrix}
2 & 1 & -1 \\
1 & 1 & -2 \\
1 & -2 & 1
\end{pmatrix}
\]

**Step 1: Find the eigenvalues of \( A \)**

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

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

Where \( I \) is the identity matrix and \( \lambda \) are the eigenvalues. 

\[
A - \lambda I = \begin{pmatrix}
2-\lambda & 1 & -1 \\
1 & 1-\lambda & -2 \\
1 & -2 & 1-\lambda
\end{pmatrix}
\]

Now, calculate the determinant of this matrix:

\[
\text{det}(A - \lambda I) = \begin{vmatrix}
2-\lambda & 1 & -1 \\
1 & 1-\lambda & -2 \\
1 & -2 & 1-\lambda
\end{vmatrix}
\]

Expanding the determinant:

\[
\text{det}(A - \lambda I) = (2-\lambda)\begin{vmatrix}
1-\lambda & -2 \\
-2 & 1-\lambda
\end{vmatrix} - 1\begin{vmatrix}
1 & -2 \\
1 & 1-\lambda
\end{vmatrix} + (-1)\begin{vmatrix}
1 & 1-\lambda \\
1 & -2
\end{vmatrix}
\]

Simplifying these determinants, we get:

\[
\text{det}(A - \lambda I) = (2-\lambda)\left[(1-\lambda)^2 - (-2)(-2)\right] - 1\left[1(1-\lambda) - (-2)\right] + (-1)\left[1(-2) - 1(1-\lambda)\right]
\]

\[
= (2-\lambda)[\lambda^2 - 2\lambda - 3] - [1-\lambda + 2] + [-2 - 1 + \lambda]
\]

Now, solving this determinant gives us the characteristic polynomial:

\[
-\lambda^3 + 4\lambda^2 - \lambda - 6 = 0
\]

Finding the roots of this cubic equation gives us the eigenvalues \(\lambda_1, \lambda_2, \lambda_3\).

**Step 2: Find the eigenvectors**

For each eigenvalue \(\lambda_i\), solve the equation:

\[
(A - \lambda_i I)\mathbf{v} = 0
\]

to find the corresponding eigenvector \(\mathbf{v}_i\).

**Step 3: Form matrices \( P \) and \( D \)**

Matrix \( P \) is formed by placing the eigenvectors as columns, and \( D \) is the diagonal matrix with the eigenvalues on the diagonal.

Finally, we have:

\[
A = PDP^{-1}
\]

Let's proceed with calculating the roots and finding the eigenvectors.

Would you like to see the detailed solution for each step?

Learn how to diagonalize the matrix A = [2 1 -1; 1 1 -2; 1 -2 1] with a detailed step-by-step solution. This problem covers eigenvalue calculation, eigenvector determination, and matrix diagonalization. Suitable for advanced high school and college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation