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calcule la matriz inversa mediante aplicacion algoritmo eliminacion de gauss jordan

To calculate the inverse of matrix \( C \) using the Gauss-Jordan elimination algorithm, we follow these steps:

Given matrix:
\[
C = \begin{pmatrix}
1 & 6 & 0 \\
5 & 3 & -6 \\
4 & -3 & 1
\end{pmatrix}
\]

### Step-by-step procedure:

1. **Augment the matrix** \( C \) with the identity matrix of the same order (3x3):

\[
\left( \begin{array}{ccc|ccc}
1 & 6 & 0 & 1 & 0 & 0 \\
5 & 3 & -6 & 0 & 1 & 0 \\
4 & -3 & 1 & 0 & 0 & 1
\end{array} \right)
\]

2. **Use row operations** to convert the left side of the augmented matrix to the identity matrix, while applying the same operations to the right side.

Let's begin performing these operations.

### Row Operations:

- **Step 1**: Make the element in the first row, first column (pivot) a 1 (which it already is), so no need to change.
  
- **Step 2**: Eliminate the elements below the pivot in column 1. We need to make the 5 and 4 in the first column of rows 2 and 3 become zeros.

   \[
   R2 \rightarrow R2 - 5R1 \quad \text{and} \quad R3 \rightarrow R3 - 4R1
   \]

   After performing these operations, the matrix becomes:

   \[
   \left( \begin{array}{ccc|ccc}
   1 & 6  & 0 & 1 & 0 & 0 \\
   0 & -27 & -6 & -5 & 1 & 0 \\
   0 & -27 & 1 & -4 & 0 & 1
   \end{array} \right)
   \]

- **Step 3**: Now, we'll simplify the second row by dividing \( R2 \) by \( -27 \) to make the pivot element in row 2 a 1:

   \[
   R2 \rightarrow \frac{R2}{-27}
   \]

   After division, we have:

   \[
   \left( \begin{array}{ccc|ccc}
   1 & 6 & 0 & 1 & 0 & 0 \\
   0 & 1 & \frac{2}{9} & \frac{5}{27} & -\frac{1}{27} & 0 \\
   0 & -27 & 1 & -4 & 0 & 1
   \end{array} \right)
   \]

- **Step 4**: Eliminate the element in the third row, second column. We perform:

   \[
   R3 \rightarrow R3 + 27R2
   \]

   The resulting matrix is:

   \[
   \left( \begin{array}{ccc|ccc}
   1 & 6 & 0 & 1 & 0 & 0 \\
   0 & 1 & \frac{2}{9} & \frac{5}{27} & -\frac{1}{27} & 0 \\
   0 & 0 & 7 & 1 & -1 & 1
   \end{array} \right)
   \]

- **Step 5**: Divide the third row by 7 to make the pivot element in the third row a 1:

   \[
   R3 \rightarrow \frac{R3}{7}
   \]

   Now the matrix is:

   \[
   \left( \begin{array}{ccc|ccc}
   1 & 6 & 0 & 1 & 0 & 0 \\
   0 & 1 & \frac{2}{9} & \frac{5}{27} & -\frac{1}{27} & 0 \\
   0 & 0 & 1 & \frac{1}{7} & -\frac{1}{7} & \frac{1}{7}
   \end{array} \right)
   \]

- **Step 6**: Eliminate the element in the first row, second column by performing:

   \[
   R1 \rightarrow R1 - 6R2
   \]

   After this operation, the matrix becomes:

   \[
   \left( \begin{array}{ccc|ccc}
   1 & 0 & -\frac{4}{3} & \frac{11}{9} & \frac{2}{9} & 0 \\
   0 & 1 & \frac{2}{9} & \frac{5}{27} & -\frac{1}{27} & 0 \\
   0 & 0 & 1 & \frac{1}{7} & -\frac{1}{7} & \frac{1}{7}
   \end{array} \right)
   \]

- **Step 7**: Finally, eliminate the element in the second row, third column by performing:

   \[
   R2 \rightarrow R2 - \frac{2}{9}R3
   \]

   The final matrix is:

   \[
   \left( \begin{array}{ccc|ccc}
   1 & 0 & 0 & \frac{25}{63} & \frac{5}{63} & \frac{2}{9} \\
   0 & 1 & 0 & \frac{11}{63} & -\frac{2}{63} & -\frac{2}{9} \\
   0 & 0 & 1 & \frac{1}{7} & -\frac{1}{7} & \frac{1}{7}
   \end{array} \right)
   \]

The right-hand side of the matrix represents the inverse of \( C \).

Thus, the inverse matrix is:
\[
C^{-1} = \begin{pmatrix}
\frac{25}{63} & \frac{5}{63} & \frac{2}{9} \\
\frac{11}{63} & -\frac{2}{63} & -\frac{2}{9} \\
\frac{1}{7} & -\frac{1}{7} & \frac{1}{7}
\end{pmatrix}
\]

Would you like more details on any of the steps? Here are five related questions you can explore:

1. What are the conditions for a matrix to have an inverse?
2. How does the determinant of a matrix affect the existence of its inverse?
3. Can you explain how row operations work in Gauss-Jordan elimination?
4. What is the difference between Gauss elimination and Gauss-Jordan elimination?
5. How can the inverse matrix be used in solving systems of equations?

**Tip**: To verify that a matrix is indeed the inverse, you can multiply the original matrix by the computed inverse and check if the product is the identity matrix.

Learn how to calculate the inverse of a 3x3 matrix using the Gauss-Jordan elimination algorithm. This step-by-step guide includes row operations and key linear algebra concepts. Suitable for students at the advanced high school or undergraduate level.

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