If a square matrix A satisfies A3 − 3A + 6I = 0, which of the following is equal to A−1 

We are given the matrix equation \( A^3 - 3A + 6I = 0 \), where \( A \) is a square matrix and \( I \) is the identity matrix.

To find \( A^{-1} \), follow these steps:

1. Rewrite the equation as:

   \[
   A^3 - 3A = -6I
   \]

   Divide both sides by \(-6\) to get:

   \[
   \frac{A^3 - 3A}{-6} = I
   \]

2. Now factor the left-hand side:

   \[
   A(A^2 - 3I) = -6I
   \]

   We want to express \( A^{-1} \) in terms of \( A \). Rewrite the equation to solve for \( A^{-1} \):

   \[
   A(A^2 - 3I) = -6I \quad \Rightarrow \quad A^{-1} = -\frac{1}{6}(A^2 - 3I)
   \]

Thus, the inverse of the matrix \( A \) is:

\[
A^{-1} = -\frac{1}{6}(A^2 - 3I)
\]

### Would you like any further details or have any questions?

Here are some related questions for further exploration:
1. How can we check if a matrix is invertible?
2. What properties must a matrix have for an inverse to exist?
3. How would you find the inverse of a 2x2 matrix by hand?
4. What is the Cayley-Hamilton theorem and how does it apply to matrices?
5. How can this matrix equation help in finding higher powers of \( A \)?

**Tip:** When dealing with matrix equations involving inverses, always ensure the matrix is non-singular (determinant is non-zero) to guarantee the existence of \( A^{-1} \).

If a square matrix A satisfies A^3 − 3A + 6I = 0, which of the following is equal to A^{-1}?

Explore the solution to finding the inverse of a square matrix A when given the equation A^3 - 3A + 6I = 0. This detailed step-by-step explanation covers key linear algebra concepts such as matrix inversion, properties of matrices, and the Cayley-Hamilton theorem. Suitable for advanced high school students and undergraduates.

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