**find inverse of matrix using gauss jordan method.**
**matrix:**
**|1  1  5|**
**|1  2  7|**
**|2  -1  4|**

To find the inverse of a matrix using the Gauss-Jordan method, we perform row operations to convert the given matrix into the identity matrix, while simultaneously performing the same operations on the identity matrix of the same size. The resulting matrix on the right will be the inverse of the given matrix.

**Matrix:**
\[
\mathbf{A} = \begin{bmatrix} 1 & 1 & 5 \\ 1 & 2 & 7 \\ 2 & -1 & 4 \end{bmatrix}
\]

**Step-by-step solution:**

1. **Augment the matrix** with the identity matrix:
   \[
   \left[\begin{array}{ccc|ccc}
   1 & 1 & 5 & 1 & 0 & 0 \\
   1 & 2 & 7 & 0 & 1 & 0 \\
   2 & -1 & 4 & 0 & 0 & 1
   \end{array}\right]
   \]

2. **Perform row operations** to convert the left side into the identity matrix. I will now perform these steps.It seems that the matrix is not invertible because a division by zero occurred during the Gauss-Jordan elimination process. This suggests that the matrix is singular, meaning its determinant is zero and thus it does not have an inverse.

### Key Insight:
- A matrix is invertible if and only if its determinant is non-zero. In this case, the determinant is zero, indicating that the matrix cannot be inverted.

Would you like more details on why the matrix is singular or need help with anything else? 

### Related Questions:
1. How to find the determinant of a matrix?
2. What makes a matrix singular or non-invertible?
3. How to perform row operations correctly in Gauss-Jordan elimination?
4. What is the significance of the identity matrix in finding inverses?
5. How can we determine if a matrix has an inverse without calculation?

**Tip:** Always check the determinant of a matrix before attempting to find its inverse, as a zero determinant indicates the matrix is not invertible.

Learn how to find the inverse of a matrix using the Gauss-Jordan method with a step-by-step solution. Understand why the matrix in question is singular and cannot be inverted due to its zero determinant. Explore related concepts such as matrix operations and the Invertible Matrix Theorem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation