https://file.aiquickdraw.com/mathbot/file/1727804263488fl171rom.jpg

The question is asking to find the matrix that does **not** have a **determinant**. In other words, we are looking for a singular matrix—one whose determinant is zero.

We can solve this by calculating the determinant of each matrix.

1. **Matrix A:**
\[
\begin{pmatrix}
15 & 2 \\
-12 & -3
\end{pmatrix}
\]
Determinant: \( \text{det}(A) = (15 \times -3) - (2 \times -12) = -45 + 24 = -21 \)

2. **Matrix B:**
\[
\begin{pmatrix}
-5 & 10 \\
-2 & -3
\end{pmatrix}
\]
Determinant: \( \text{det}(B) = (-5 \times -3) - (10 \times -2) = 15 + 20 = 35 \)

3. **Matrix C:**
\[
\begin{pmatrix}
-3 & -6 \\
-6 & -12
\end{pmatrix}
\]
Determinant: \( \text{det}(C) = (-3 \times -12) - (-6 \times -6) = 36 - 36 = 0 \)

4. **Matrix D:**
\[
\begin{pmatrix}
9 & -6 \\
12 & 8
\end{pmatrix}
\]
Determinant: \( \text{det}(D) = (9 \times 8) - (-6 \times 12) = 72 + 72 = 144 \)

### Conclusion:
Matrix **C** has a determinant of 0, making it singular. Therefore, matrix **C** does not have an inverse.

Do you want more details or have any questions?

Here are 5 related questions:
1. How do you find the determinant of a 3x3 matrix?
2. What is the significance of a matrix having a determinant of 0?
3. How can you verify if a matrix is invertible?
4. What are the properties of a singular matrix?
5. How does row reduction relate to determining a matrix's singularity?

**Tip:** A square matrix with linearly dependent rows or columns will always have a determinant of zero.

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Learn how to identify the singular matrix by calculating the determinant of 2x2 matrices. This problem involves calculating determinants using the formula det(A) = ad - bc and recognizing when a matrix does not have an inverse. Suitable for high school students.

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