Given the matrix M = [−12330121−1].
a. Calculate the determinant of M. (Show your work!)
b. Using the Gauss-Jordan algorithm, calculate the inverse of M.
c. Using M-1, solve the following system of equations.
-x + 2y + 3z = -9,
2x + y – z = -3,
3x + z = 13.

### Given Matrix \( M \):
\[
M = \begin{pmatrix} -1 & 2 & 3 \\ 0 & 1 & 2 \\ 1 & -1 & -1 \end{pmatrix}
\]

Let's go through the parts step-by-step:

### Part (a): Calculate the Determinant of \( M \)

The determinant of a 3x3 matrix:
\[
M = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}
\]
is given by:
\[
\text{det}(M) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

For the matrix \( M \), we have:
\[
M = \begin{pmatrix} -1 & 2 & 3 \\ 0 & 1 & 2 \\ 1 & -1 & -1 \end{pmatrix}
\]
So, the determinant is:
\[
\text{det}(M) = -1 \cdot (1 \cdot -1 - 2 \cdot -1) - 2 \cdot (0 \cdot -1 - 2 \cdot 1) + 3 \cdot (0 \cdot -1 - 1 \cdot 1)
\]

Let me calculate this determinant now.It seems my tool access has been reset. However, I can guide you through calculating the determinant manually:

### Part (a) Manual Calculation of the Determinant

Given the matrix \( M = \begin{pmatrix} -1 & 2 & 3 \\ 0 & 1 & 2 \\ 1 & -1 & -1 \end{pmatrix} \), the determinant is calculated as:

\[
\text{det}(M) = -1 \cdot ((1 \times -1) - (2 \times -1)) - 2 \cdot ((0 \times -1) - (2 \times 1)) + 3 \cdot ((0 \times -1) - (1 \times 1))
\]

Now, calculate each term:

1. \( -1 \cdot ((1 \times -1) - (2 \times -1)) = -1 \cdot (-1 + 2) = -1 \cdot 1 = -1 \)
2. \( -2 \cdot ((0 \times -1) - (2 \times 1)) = -2 \cdot (0 - 2) = -2 \cdot -2 = 4 \)
3. \( 3 \cdot ((0 \times -1) - (1 \times 1)) = 3 \cdot (0 - 1) = 3 \cdot -1 = -3 \)

Summing these:
\[
\text{det}(M) = -1 + 4 - 3 = 0
\]

Thus, the determinant of \( M \) is **0**.

### Conclusion

Since the determinant is 0, the matrix \( M \) is **singular** and does not have an inverse. Therefore:

- **Part (b)**: The inverse of \( M \) cannot be calculated because the matrix is singular (non-invertible).
- **Part (c)**: Since there is no inverse, we cannot solve the system of equations using the matrix inversion method.

If you have any questions or need further clarification, feel free to ask!

Here are 5 related questions:
1. What does it mean for a matrix to be singular?
2. How is the determinant related to the invertibility of a matrix?
3. Can we still solve the system of equations using other methods?
4. How would you use Gaussian elimination to solve a system without using matrix inversion?
5. What happens if the determinant of a matrix is close to zero but not exactly zero?

**Tip:** For solving linear systems, even when a matrix is singular, methods like Gaussian elimination can sometimes reveal dependent equations, indicating multiple or no solutions.

Learn how to calculate the determinant and inverse of a 3x3 matrix using Gauss-Jordan elimination. Includes step-by-step solutions and explanations for solving a system of linear equations using matrix inversion. Suitable for undergraduate level study.

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