https://file.aiquickdraw.com/mathbot/file/1729617539072mzzdxkvk.jpg

The given problem asks you to verify the **Cayley-Hamilton Theorem** for the matrix \( A \), and then find both \( A^{-1} \) (the inverse of \( A \)) and \( A^4 \) (the fourth power of \( A \)).

### Step 1: Verify the Cayley-Hamilton Theorem
The **Cayley-Hamilton Theorem** states that a square matrix satisfies its own characteristic equation.

Given:
\[
A = \begin{pmatrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 1 & -1 & 1 \end{pmatrix}
\]

#### Step 1.1: Find the characteristic polynomial
The characteristic polynomial of a matrix \( A \) is given by:
\[
\text{det}(A - \lambda I) = 0
\]
where \( \lambda \) is the eigenvalue, and \( I \) is the identity matrix.

So, calculate \( A - \lambda I \):
\[
A - \lambda I = \begin{pmatrix} 1 - \lambda & 0 & 3 \\ 2 & 1 - \lambda & -1 \\ 1 & -1 & 1 - \lambda \end{pmatrix}
\]

Now, find the determinant of this matrix:
\[
\text{det}(A - \lambda I) = \begin{vmatrix} 1 - \lambda & 0 & 3 \\ 2 & 1 - \lambda & -1 \\ 1 & -1 & 1 - \lambda \end{vmatrix}
\]

Expanding along the first row:
\[
\text{det}(A - \lambda I) = (1 - \lambda) \begin{vmatrix} 1 - \lambda & -1 \\ -1 & 1 - \lambda \end{vmatrix} - 0 + 3 \begin{vmatrix} 2 & 1 - \lambda \\ 1 & -1 \end{vmatrix}
\]

First, simplify the smaller 2x2 determinants:
\[
\begin{vmatrix} 1 - \lambda & -1 \\ -1 & 1 - \lambda \end{vmatrix} = (1 - \lambda)(1 - \lambda) - (-1)(-1) = (1 - \lambda)^2 - 1 = \lambda^2 - 2\lambda
\]
\[
\begin{vmatrix} 2 & 1 - \lambda \\ 1 & -1 \end{vmatrix} = 2(-1) - (1 - \lambda)(1) = -2 - (1 - \lambda) = -2 - 1 + \lambda = \lambda - 3
\]

Now substitute back into the determinant equation:
\[
\text{det}(A - \lambda I) = (1 - \lambda)(\lambda^2 - 2\lambda) + 3(\lambda - 3)
\]
Simplify:
\[
= (1 - \lambda)(\lambda^2 - 2\lambda) + 3\lambda - 9
\]
\[
= \lambda^2 - 2\lambda^3 - \lambda^2 + 2\lambda^2 + 3\lambda - 9
\]
\[
= -\lambda^3 + 2\lambda^2 + 3\lambda - 9
\]

So the characteristic polynomial is:
\[
p(\lambda) = -\lambda^3 + 2\lambda^2 + 3\lambda - 9
\]

### Step 1.2: Apply the Cayley-Hamilton Theorem
According to the Cayley-Hamilton Theorem, the matrix \( A \) satisfies its own characteristic polynomial:
\[
A^3 - 2A^2 - 3A + 9I = 0
\]
We will use this equation to compute powers of \( A \) and the inverse.

### Step 2: Find \( A^{-1} \)
Once we verify the Cayley-Hamilton Theorem, we can find the inverse of \( A \) by expressing it using the characteristic polynomial.

### Step 3: Find \( A^4 \)
Using the Cayley-Hamilton equation, \( A^4 \) can be expressed in terms of lower powers of \( A \).

---

Would you like more details on finding \( A^{-1} \) and \( A^4 \), or any specific part of the process explained further?

Here are 5 related questions for better understanding:
1. What are eigenvalues, and how do they relate to the characteristic polynomial?
2. Can the Cayley-Hamilton Theorem be applied to non-square matrices?
3. How can powers of matrices be computed more efficiently using the Cayley-Hamilton Theorem?
4. What methods are available for finding the inverse of a matrix other than using the characteristic polynomial?
5. How does the determinant of a matrix help in determining if the matrix is invertible?

**Tip:** The Cayley-Hamilton Theorem is a powerful tool to simplify matrix computations, especially for higher powers of matrices.

Verify Cayley-Hamilton Theorem for matrix A and determine A⁻¹ and A⁴.

Step-by-step solution for verifying the Cayley-Hamilton Theorem for a 3x3 matrix, followed by the calculation of the matrix inverse A⁻¹ and the fourth power A⁴. This problem explores advanced linear algebra concepts such as matrix powers and characteristic polynomials.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation