Find the values of the following determinants
`|{:(-3,-2),(1,5):}|`

Answer

To find the value of the determinant given by the matrix:

\[ \begin{pmatrix} -3 & -2 \\ 1 & 5 \end{pmatrix} \]

we can use the formula for the determinant of a 2x2 matrix, which is given by:

\[ \text{det}(A) = a_{11} \cdot a_{22} - a_{12} \cdot a_{21} \]

where:
- \( a_{11} \) is the element in the first row and first column,
- \( a_{12} \) is the element in the first row and second column,
- \( a_{21} \) is the element in the second row and first column,
- \( a_{22} \) is the element in the second row and second column.

### Step-by-step Solution:

1. **Identify the elements of the matrix:**
- \( a_{11} = -3 \)
- \( a_{12} = -2 \)
- \( a_{21} = 1 \)
- \( a_{22} = 5 \)

2. **Apply the determinant formula:**
\[ \text{det}(A) = (-3) \cdot (5) - (-2) \cdot (1) \]

3. **Calculate the products:**
- First product: \( -3 \cdot 5 = -15 \)
- Second product: \( -2 \cdot 1 = -2 \)

4. **Substitute the products back into the formula:**
\[ \text{det}(A) = -15 - (-2) \]

5. **Simplify the expression:**
\[ \text{det}(A) = -15 + 2 = -13 \]

Thus, the value of the determinant is:

\[ \text{det}(A) = -13 \]