https://r2.erweima.ai/mathbot/image/172682365315734lkfg6c.jpg

Your work appears to be a matrix computation involving the steps to calculate the inverse of a matrix, using row reduction techniques. Let's cross-check the process and result:

### Steps Outline:
1. **Matrix Setup:**  
   You start with a matrix \( A \) of size \( 3 \times 3 \):
   \[
   A = \begin{bmatrix}
   0 & 2 & 3 \\
   2 & 4 & 0 \\
   3 & 0 & 1
   \end{bmatrix}
   \]
   The identity matrix \( I \) is augmented to \( A \):
   \[
   \left[ A | I \right] = \left[ \begin{bmatrix}
   0 & 2 & 3 \\
   2 & 4 & 0 \\
   3 & 0 & 1
   \end{bmatrix} \middle| \begin{bmatrix}
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & 0 & 1
   \end{bmatrix} \right]
   \]

2. **Row operations:**  
   You use row reduction to bring \( A \) to row echelon form (reduce it to \( I \)) and apply the same row operations to the identity matrix \( I \). The goal is to transform \( A \) into \( I \), leaving the transformed \( I \) as \( A^{-1} \).

   Key row operations visible in your steps include:
   - Swapping rows \( R_1 \) and \( R_3 \) since \( R_1 \) starts with 0.
   - Various elementary row operations to achieve zeros and ones where needed.

3. **Final Matrix Inverse:**
   According to your solution, the inverse of \( A \) is given as:
   \[
   A^{-1} = \begin{bmatrix}
   -4 & 4 & 12 \\
   4/3 & 8/3 & 9/3 \\
   12 & -12 & -4
   \end{bmatrix}
   \]

---

### Cross-checking:

I'll verify if the steps and the result you reached for the inverse matrix are correct by redoing the row operations and checking the determinant. Let me calculate that now.The inverse matrix of \( A \), calculated using numpy, is:

\[
A^{-1} = \begin{bmatrix}
-0.1 & 0.05 & 0.3 \\
0.05 & 0.225 & -0.15 \\
0.3 & -0.15 & 0.1
\end{bmatrix}
\]

This doesn't match the matrix you derived in your notebook, which was:

\[
A^{-1} = \begin{bmatrix}
-4 & 4 & 12 \\
4/3 & 8/3 & 9/3 \\
12 & -12 & -4
\end{bmatrix}
\]

It seems there is a mistake in the row reduction process. The values differ from the correct inverse.

### Suggestions for Cross-Check:
- Double-check each elementary row operation step.
- Ensure the row-reduction follows Gaussian elimination rules, and each operation is applied correctly.

Would you like further details on the individual steps involved in the row reduction or an explanation of the process?

---

Here are 5 related questions to deepen understanding:
1. How is the determinant of a matrix related to its invertibility?
2. Can a matrix with a row of zeros have an inverse? Why or why not?
3. What is the significance of row-reduction in matrix operations?
4. How does matrix inversion apply to solving systems of linear equations?
5. What are some efficient methods for computing matrix inverses in higher dimensions?

**Tip:** Always check the determinant of a matrix before attempting to invert it, as a zero determinant means the matrix is singular and non-invertible.

Cross-checking a user's matrix inversion process using Gaussian elimination. This solution involves step-by-step row reduction to compute the inverse of a 3x3 matrix, following elementary row operations. Ideal for students studying Linear Algebra at the undergraduate level.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation