To find the determinant of the [tex]\( 2 \times 2 \)[/tex] matrix
[tex]\[
\left[\begin{array}{cc}
-7 & 1 \\
-5 & 11
\end{array}\right],
\][/tex]
we use the formula for the determinant of a [tex]\( 2 \times 2 \)[/tex] matrix. For a general matrix
[tex]\[
\left[\begin{array}{cc}
a & b \\
c & d
\end{array}\right],
\][/tex]
the determinant [tex]\(\det\)[/tex] is given by:
[tex]\[
\det = ad - bc.
\][/tex]
In this specific case:
- [tex]\( a = -7 \)[/tex]
- [tex]\( b = 1 \)[/tex]
- [tex]\( c = -5 \)[/tex]
- [tex]\( d = 11 \)[/tex]
Substitute these values into the determinant formula:
[tex]\[
\det = (-7)(11) - (1)(-5).
\][/tex]
Now, let's calculate the products:
[tex]\[
(-7)(11) = -77
\][/tex]
and
[tex]\[
(1)(-5) = -5.
\][/tex]
Next, combine these results according to the determinant formula:
[tex]\[
\det = -77 - (-5).
\][/tex]
Subtracting a negative is the same as adding the positive equivalent, so we get:
[tex]\[
\det = -77 + 5.
\][/tex]
Finally, perform the addition:
[tex]\[
\det = -72.
\][/tex]
Thus, the determinant of the matrix is:
[tex]\[
-72.
\][/tex]
