To find the determinant of the given 2x2 matrix, we can use the formula for the determinant of a 2x2 matrix. The formula is:
[tex]\[
\text{det} \left| \begin{array}{cc}
a & b \\
c & d
\end{array} \right|
= ad - bc
\][/tex]
Given the matrix:
[tex]\[
\left| \begin{array}{cc}
-7 & 2 \\
4 & 5
\end{array} \right|
\][/tex]
we identify the elements of the matrix as:
- [tex]\(a = -7\)[/tex],
- [tex]\(b = 2\)[/tex],
- [tex]\(c = 4\)[/tex],
- [tex]\(d = 5\)[/tex].
Using the determinant formula, we substitute these values into the expression:
[tex]\[
\text{det} \left| \begin{array}{cc}
-7 & 2 \\
4 & 5
\end{array} \right|
= (-7 \times 5) - (2 \times 4)
\][/tex]
Perform the multiplications inside the parentheses first:
[tex]\[
(-7 \times 5) = -35
\][/tex]
[tex]\[
(2 \times 4) = 8
\][/tex]
Next, subtract the second product from the first:
[tex]\[
-35 - 8 = -43
\][/tex]
Thus, the determinant of the given matrix is:
[tex]\[
\left|\begin{array}{cc}-7 & 2 \\ 4 & 5\end{array}\right| = -43
\][/tex]
