To evaluate the determinant of the given [tex]\( 2 \times 2 \)[/tex] matrix:
[tex]\[
\left|\begin{array}{rr}
1 & 3 \\
-1 & -3
\end{array}\right|
\][/tex]
we use the formula for the determinant of a [tex]\( 2 \times 2 \)[/tex] matrix of the form:
[tex]\[
\left|\begin{array}{cc}
a & b \\
c & d
\end{array}\right| = ad - bc
\][/tex]
For our given matrix, we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 3 \)[/tex]
- [tex]\( c = -1 \)[/tex]
- [tex]\( d = -3 \)[/tex]
By substituting these values into the formula:
[tex]\[
\text{Determinant} = (1 \cdot -3) - (3 \cdot -1)
\][/tex]
Calculating inside the parentheses:
[tex]\[
\text{Determinant} = -3 - (-3)
\][/tex]
Simplifying the expression inside the parentheses:
[tex]\[
\text{Determinant} = -3 + 3
\][/tex]
Finally, adding the values:
[tex]\[
\text{Determinant} = 0
\][/tex]
Therefore, the determinant of the given matrix is [tex]\( 0 \)[/tex].
