5. If lambda ^ 3 - 6lambda ^ 2 + 9lambda = 4 is the characteristic equation of a square matrix A, then A ^ - 1 is equal to

Given that \(\lambda^3 - 6\lambda^2 + 9\lambda = 4\) is the characteristic equation of a square matrix \(A\), we need to find the inverse of matrix \(A\), denoted as \(A^{-1}\).

### Step 1: Express the characteristic equation in standard form
The standard form of the characteristic equation is:
\[
\lambda^3 - 6\lambda^2 + 9\lambda - 4 = 0
\]

### Step 2: Use the Cayley-Hamilton Theorem
The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. So, for the matrix \(A\), the equation becomes:
\[
A^3 - 6A^2 + 9A - 4I = 0
\]
where \(I\) is the identity matrix of the same order as \(A\).

### Step 3: Solve for \(A^{-1}\)
To find \(A^{-1}\), we need to manipulate the equation \(A^3 - 6A^2 + 9A - 4I = 0\) to express \(I\) (the identity matrix) in terms of \(A\):
\[
A^3 - 6A^2 + 9A = 4I
\]
Now, divide both sides by 4:
\[
\frac{A^3 - 6A^2 + 9A}{4} = I
\]
Thus, the inverse of \(A\) is given by:
\[
A^{-1} = \frac{A^3 - 6A^2 + 9A}{4}
\]

### Conclusion
\[
A^{-1} = \frac{1}{4}(A^3 - 6A^2 + 9A)
\]

This expression represents the inverse of the matrix \(A\).

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Would you like more details or have any questions? Here are some related questions:

1. How is the Cayley-Hamilton theorem used to find the inverse of a matrix?
2. What is the significance of the characteristic equation in matrix theory?
3. Can you find the determinant of \(A\) using the given characteristic equation?
4. How can you verify the inverse of a matrix found using the characteristic equation?
5. What are the general properties of the characteristic equation for a 3x3 matrix?

**Tip:** When finding the inverse of a matrix using its characteristic equation, always ensure that the matrix is non-singular, i.e., its determinant is non-zero.

Discover how to find the inverse of matrix A using the characteristic equation λ³ - 6λ² + 9λ = 4 with detailed steps and explanations. This problem involves matrix theory, the Cayley-Hamilton theorem, and linear algebra concepts. Suitable for college-level students and anyone interested in matrix algebra.

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