To find the determinant of a 2x2 matrix, we can use a straightforward formula. For any 2x2 matrix of the form:
[tex]\[
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\][/tex]
The determinant is calculated as:
[tex]\[
\text{det} = (a \cdot d) - (b \cdot c)
\][/tex]
Given the matrix:
[tex]\[
\begin{bmatrix}
-2 & 5 \\
1 & 4
\end{bmatrix}
\][/tex]
We can identify the elements as follows:
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = 5 \)[/tex]
- [tex]\( c = 1 \)[/tex]
- [tex]\( d = 4 \)[/tex]
Now we substitute these values into the determinant formula:
[tex]\[
\text{det} = (-2 \cdot 4) - (5 \cdot 1)
\][/tex]
First, calculate the products:
[tex]\[
-2 \cdot 4 = -8
\][/tex]
[tex]\[
5 \cdot 1 = 5
\][/tex]
Then, subtract the second product from the first:
[tex]\[
-8 - 5 = -13
\][/tex]
Therefore, the determinant of the matrix is:
[tex]\[
\boxed{-13}
\][/tex]
