Using Cayley Hamilton theorem, find the inverse of  . Also compute A4.


To find the inverse of matrix \( A \) and compute \( A^4 \) using the Cayley-Hamilton theorem, we proceed as follows:

1. **Cayley-Hamilton Theorem**: According to this theorem, every square matrix satisfies its own characteristic polynomial. If \( A \) is an \( n \times n \) matrix with characteristic polynomial \( p(\lambda) \), then \( p(A) = 0 \).

### Step 1: Find the Characteristic Polynomial of \( A \)
Let \( A \) be the given matrix. To find the characteristic polynomial, we compute the determinant of \( A - \lambda I \), where \( I \) is the identity matrix and \( \lambda \) is a scalar:
\[
\text{det}(A - \lambda I) = 0
\]
This will yield the characteristic polynomial \( p(\lambda) \).

### Step 2: Apply Cayley-Hamilton Theorem
Substitute \( A \) into its characteristic polynomial to get a matrix equation that \( A \) satisfies. This equation will be of the form:
\[
A^n + c_{n-1}A^{n-1} + \cdots + c_1A + c_0I = 0
\]
where \( c_0, c_1, \dots, c_{n-1} \) are constants derived from the characteristic polynomial.

### Step 3: Solve for \( A^{-1} \)
Rearrange the matrix equation to isolate terms involving \( A \) on one side and solve for \( A^{-1} \) in terms of powers of \( A \).

### Step 4: Compute \( A^4 \)
Using the relation from the Cayley-Hamilton theorem, express \( A^4 \) in terms of lower powers of \( A \).

To proceed, please provide the actual entries of matrix \( A \) so I can compute these values specifically.

Using Cayley Hamilton theorem, find the inverse of A. Also compute A^4.

Explore how to find the inverse of a matrix and compute A^4 using the Cayley-Hamilton theorem. This solution demonstrates characteristic polynomial derivation and matrix inversion, essential in linear algebra. Perfect for undergraduate students studying matrix algebra.

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