https://file.aiquickdraw.com/mathbot/file/1725905035612fuzz8e3a.jpg

We are given the matrix equation:

\[
\begin{pmatrix}
3 & -6 \\
-1 & 2 \\
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
\end{pmatrix}
=
\begin{pmatrix}
4 \\
9 \\
\end{pmatrix}
\]

We can solve this system by determining whether the coefficient matrix has an inverse or not. If it has an inverse, we can solve the system uniquely. If the matrix does not have an inverse, there is no unique solution.

### Step 1: Check if the coefficient matrix has an inverse
To determine if the matrix is invertible, we need to calculate its **determinant**.

\[
\text{Determinant of } \begin{pmatrix} 3 & -6 \\ -1 & 2 \end{pmatrix} = (3)(2) - (-6)(-1)
\]
\[
= 6 - 6 = 0
\]

Since the determinant is **0**, the matrix is **not invertible**, meaning that there is **no unique solution** to this system of equations.

### Conclusion:
The matrix has no inverse, so the system has no unique solution.

Let me know if you would like more details on this or have any questions!

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Here are five questions for further exploration:

1. What does it mean for a matrix to have no inverse?
2. How can we interpret the solution set of a matrix system with no unique solution?
3. What is the difference between consistent and inconsistent systems in linear algebra?
4. How can we solve a system when the coefficient matrix is singular (non-invertible)?
5. What is the significance of the determinant being zero for a matrix?

**Tip:** A matrix is invertible if and only if its determinant is non-zero. If the determinant is zero, the matrix is said to be singular, and the system may have infinitely many solutions or none at all.

Solve the matrix equation. If the coefficient matrix has no inverse, write no unique solution.

This solution explores how to solve a matrix equation by determining the invertibility of the coefficient matrix using the determinant. If the determinant is zero, the matrix is non-invertible, and the system has no unique solution. Ideal for college-level linear algebra students.

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