To find the determinant of a [tex]\( 2 \times 2 \)[/tex] matrix, we use the formula:
[tex]\[
\text{det}\left(\begin{array}{cc}
a & b \\
c & d
\end{array}\right) = ad - bc
\][/tex]
Given the matrix:
[tex]\[
\left[\begin{array}{ll}
6 & -1 \\
2 & -9
\end{array}\right],
\][/tex]
we identify [tex]\( a = 6 \)[/tex], [tex]\( b = -1 \)[/tex], [tex]\( c = 2 \)[/tex], and [tex]\( d = -9 \)[/tex].
Substituting these values into the determinant formula, we get:
[tex]\[
\text{det} = (6 \cdot -9) - (-1 \cdot 2)
\][/tex]
Calculate each term separately:
[tex]\[
6 \cdot -9 = -54
\][/tex]
[tex]\[
-1 \cdot 2 = -2
\][/tex]
Combine these results:
[tex]\[
\text{det} = -54 - (-2) = -54 + 2 = -52
\][/tex]
Therefore, the determinant of the matrix is:
[tex]\[
\text{det} = -52
\][/tex]
