To determine which expression correctly represents the determinant of the matrix [tex]\( A = \left[\begin{array}{ll}2 & 3 \\ 1 & 9\end{array}\right] \)[/tex], we will follow the formula for calculating the determinant of a 2x2 matrix.
For a general 2x2 matrix:
[tex]\[ A = \left[\begin{array}{ll}a & b \\ c & d\end{array}\right], \][/tex]
the determinant of [tex]\( A \)[/tex] is given by:
[tex]\[ \det(A) = ad - bc. \][/tex]
Applying this formula to our specific matrix:
[tex]\[ A = \left[\begin{array}{ll}2 & 3 \\ 1 & 9\end{array}\right], \][/tex]
we identify [tex]\( a = 2 \)[/tex], [tex]\( b = 3 \)[/tex], [tex]\( c = 1 \)[/tex], and [tex]\( d = 9 \)[/tex].
Plugging these values into the determinant formula, we get:
[tex]\[ \det(A) = (2)(9) - (1)(3). \][/tex]
Therefore, the correct expression for the determinant of [tex]\( A \)[/tex] is:
[tex]\[ \det(A) = (2)(9) - (1)(3). \][/tex]
Among the given options:
1. [tex]\(\operatorname{det}(A) = (2)(9) - (1)(3)\)[/tex]
2. [tex]\(\operatorname{det}(A) = (2)(9) + (1)(3)\)[/tex]
3. [tex]\(\operatorname{det}(A) = (2)(3) - (1)(9)\)[/tex]
4. [tex]\(\operatorname{det}(A) = (2)(3) + (1)(9)\)[/tex]
The first option, [tex]\(\operatorname{det}(A) = (2)(9) - (1)(3)\)[/tex], is the correct one.
