To find the determinant of the matrix [tex]\( C = \begin{pmatrix} 3 & -2 \\ -5 & -7 \end{pmatrix} \)[/tex], we will use the formula for the determinant of a 2x2 matrix. The formula is given by:
[tex]\[
\text{det}(C) = ad - bc
\][/tex]
where [tex]\( C = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \)[/tex].
For the given matrix [tex]\( C \)[/tex]:
[tex]\[
a = 3, \quad b = -2, \quad c = -5, \quad d = -7
\][/tex]
Plugging these values into the formula, we get:
[tex]\[
\text{det}(C) = (3 \cdot -7) - (-2 \cdot -5)
\][/tex]
First, compute the product [tex]\( 3 \cdot -7 \)[/tex]:
[tex]\[
3 \cdot -7 = -21
\][/tex]
Next, compute the product [tex]\(-2 \cdot -5\)[/tex]:
[tex]\[
-2 \cdot -5 = 10
\][/tex]
Now, subtract the second product from the first:
[tex]\[
\text{det}(C) = -21 - 10 = -31
\][/tex]
Thus, the determinant of matrix [tex]\( C \)[/tex] is [tex]\( -31 \)[/tex].
Among the given options, the correct answer is:
[tex]\[
\boxed{-31}
\][/tex]
