https://file.aiquickdraw.com/mathbot/file/1726421787013m8rcnyhw.jpg

The image presents two matrices, and you're asked to find their determinants using properties. Let’s address each matrix:

### Matrix (a):
\[
\begin{vmatrix}
1 & 4 & 2 \\
0 & 0 & 0 \\
3 & 9 & 5
\end{vmatrix}
\]

**Step 1: Analyze the matrix**  
Since the second row consists entirely of zeros, **the determinant of this matrix is zero**. This is a standard property: if any row or column of a matrix is entirely composed of zeros, the determinant of the matrix is zero.

So, for matrix (a):
\[
\boxed{0}
\]

### Matrix (b):
\[
\begin{vmatrix}
1 & 4 & 2 \\
1 & 4 & 2 \\
3 & 9 & 5
\end{vmatrix}
\]

**Step 1: Row duplication**  
Here, rows 1 and 2 are identical. According to the properties of determinants, if two rows (or columns) of a matrix are identical, the determinant is **zero**.

So, for matrix (b):
\[
\boxed{0}
\]

Would you like further details or explanations?

### Follow-up Questions:
1. How do row operations affect the determinant?
2. What happens to the determinant when two rows are swapped?
3. Can you calculate the determinant of a triangular matrix easily?
4. What are the properties of determinants when adding multiples of rows?
5. How does scaling a row by a scalar affect the determinant?

### Tip:
Always check for row or column simplifications (zeros or identical rows/columns) to quickly determine if the determinant is zero!

Find the determinant of two matrices using properties. The matrices are: 
Matrix (a):
| 1 4 2 |
| 0 0 0 |
| 3 9 5 |
Matrix (b):
| 1 4 2 |
| 1 4 2 |
| 3 9 5 |

Learn how to find the determinant of matrices using key properties. This problem covers important matrix properties, such as zero rows and identical rows, with a step-by-step solution. Suitable for students in Grades 10-12.

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