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The given matrix \( M \) is:

\[
M = \begin{pmatrix}
1 & 1 & 0 \\
3 & 1 & 1 \\
0 & 1 & -1
\end{pmatrix}
\]

We are asked to find the inverse of the matrix using the **Gauss-Jordan method**, which involves performing row operations to transform the matrix into the identity matrix, while applying the same operations to an identity matrix of the same size. The augmented matrix would be:

\[
\left( M | I \right) = \left( \begin{array}{ccc|ccc}
1 & 1 & 0 & 1 & 0 & 0 \\
3 & 1 & 1 & 0 & 1 & 0 \\
0 & 1 & -1 & 0 & 0 & 1
\end{array} \right)
\]

I'll perform the Gauss-Jordan elimination on this augmented matrix step by step to find the inverse.

Let me calculate that.The inverse of the matrix \( M \) is:

\[
M^{-1} = \begin{pmatrix}
-2 & 1 & 1 \\
3 & -1 & -1 \\
3 & -1 & -2
\end{pmatrix}
\]

This was computed using the Gauss-Jordan elimination method.

Would you like further details on the steps involved, or have any specific questions?

Here are five related questions for further exploration:
1. How is the Gauss-Jordan method different from Gaussian elimination?
2. What conditions must a matrix meet to have an inverse?
3. Can you explain why the identity matrix is used during the Gauss-Jordan elimination process?
4. How would you handle a non-invertible (singular) matrix?
5. How can matrix inversion be applied in solving systems of linear equations?

**Tip:** Always check the determinant of a matrix before attempting to find its inverse. If the determinant is zero, the matrix is singular and does not have an inverse.

Use the Gauss-Jordan method to compute the inverse, if it exists, of the matrix M = [ [1, 1, 0], [3, 1, 1], [0, 1, -1] ].

Learn how to compute the inverse of a 3x3 matrix using the Gauss-Jordan method. This step-by-step guide demonstrates how to transform the matrix into the identity matrix and perform the corresponding operations on an identity matrix to find the inverse. Ideal for undergraduate students studying linear algebra.

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