https://file.aiquickdraw.com/mathbot/file/1725563495078qardyd0i.jpg

The question asks to find \( A^{-1} \) using the **Cayley-Hamilton theorem** for the given matrix \( A \):

\[
A = \begin{pmatrix}
2 & 0 & 1 \\
0 & 2 & -1 \\
1 & -1 & 2
\end{pmatrix}
\]

The **Cayley-Hamilton theorem** states that every square matrix satisfies its own characteristic equation. Here's the step-by-step approach:

1. **Find the characteristic polynomial** of \( A \), which is given by:

   \[
   p_A(\lambda) = \text{det}(A - \lambda I)
   \]
   
   where \( I \) is the identity matrix and \( \lambda \) is a scalar.

2. **Substitute** the matrix \( A \) into its characteristic polynomial to get the matrix equation.
   
3. From this matrix equation, you will express \( A^{-1} \) in terms of powers of \( A \).

I'll calculate this now.The characteristic polynomial of matrix \( A \) is:

\[
p_A(\lambda) = -\lambda^3 + 6\lambda^2 - 10\lambda + 4
\]

According to the **Cayley-Hamilton theorem**, the matrix \( A \) satisfies this characteristic equation. Therefore, we substitute \( A \) for \( \lambda \) in the polynomial:

\[
A^3 - 6A^2 + 10A - 4I = 0
\]

From this equation, we can express \( A^{-1} \). I'll now solve for \( A^{-1} \).The inverse of the matrix \( A \), using the Cayley-Hamilton theorem, is:

\[
A^{-1} = \frac{1}{4} \begin{pmatrix}
12 & 2 & -6 \\
2 & 8 & 2 \\
-6 & 2 & 12
\end{pmatrix}
\]

Would you like more details on any of the steps?

Here are five related questions that expand on this concept:

1. What is the significance of the Cayley-Hamilton theorem in linear algebra?
2. How would you compute the inverse of a matrix using a direct method like Gaussian elimination?
3. Can the Cayley-Hamilton theorem be applied to matrices of all sizes?
4. What is the difference between the characteristic polynomial and the minimal polynomial of a matrix?
5. How can you use eigenvalues to find the determinant and trace of a matrix?

**Tip:** When using the Cayley-Hamilton theorem, the matrix powers in the characteristic polynomial allow you to compute functions of matrices efficiently, without directly needing their inverses.

Using the Cayley-Hamilton theorem to find A^{-1} where A is a 3x3 matrix:
A = [ [2, 0, 1], [0, 2, -1], [1, -1, 2] ]

Learn how to use the Cayley-Hamilton theorem to compute the inverse of the 3x3 matrix A = [[2, 0, 1], [0, 2, -1], [1, -1, 2]]. This linear algebra problem covers matrix inversion, the characteristic polynomial, and solving for A^{-1}. Suitable for undergraduate-level 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