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The given problem asks you to show that \( A A^{-1} = I \), where \( A \) is the matrix:

\[
A = \begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 1 & 1 & 1 \end{bmatrix}
\]

In this case, we need to demonstrate that multiplying matrix \( A \) by its inverse \( A^{-1} \) results in the identity matrix \( I \), which is a square matrix with 1's on the diagonal and 0's elsewhere.

To proceed, we will follow these steps:
1. **Find the determinant** of matrix \( A \). The determinant must be non-zero for \( A \) to have an inverse.
2. **Compute the inverse** of matrix \( A \).
3. **Multiply \( A \) by its inverse** to check if the result is the identity matrix.

Let's start by calculating the determinant of \( A \). I'll compute that now.The determinant of matrix \( A \) is \( -7 \), which is non-zero. This means that matrix \( A \) is invertible, and we can proceed to compute its inverse.

Now, let's find the inverse of matrix \( A \).It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, you can manually compute the inverse of the matrix \( A \) using these steps:

1. **Adjugate of A**: Calculate the matrix of cofactors, then transpose it.
2. **Inverse of A**: Use the formula \( A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) \).

Once you have the inverse, multiply it by \( A \) to verify whether it results in the identity matrix.

Would you like a more detailed explanation of how to compute the inverse step by step?

Here are 5 questions you might find interesting:
1. How do you calculate the determinant of a 3x3 matrix?
2. What is the cofactor matrix, and how is it used in finding the inverse?
3. Can every square matrix be inverted? Why or why not?
4. What are some properties of the identity matrix?
5. How does the adjugate differ from the transpose of a matrix?

**Tip**: Always check the determinant before trying to invert a matrix! If it's zero, the matrix has no inverse.

Let A = [ [3, -3, 4], [2, -3, 4], [1, 1, 1] ]. Show that A*A^(-1) is an identity matrix.

In this problem, we explore how to prove that multiplying matrix A by its inverse results in an identity matrix. Learn how to compute the determinant, find the inverse of a 3x3 matrix, and verify matrix multiplication.

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