To discern the correct formula for the inverse of a 2x2 matrix, let’s first recall the general formula for finding the inverse of a matrix. If we have a matrix [tex]\( A \)[/tex] given by:
[tex]\[ A = \left[\begin{array}{cc} a & b \\ c & d \end{array}\right] \][/tex]
The inverse of [tex]\( A \)[/tex], denoted as [tex]\( A^{-1} \)[/tex], can be found using the formula:
[tex]\[ A^{-1} = \frac{1}{ad - bc} \left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right] \][/tex]
Now let's verify the option given:
Option a:
[tex]\[
\frac{1}{ab - dc} \left[\begin{array}{lr} -d & b \\ c & -a \end{array}\right]
\][/tex]
Here, the denominator should be [tex]\( ad - bc \)[/tex], but in the given option, it is written as [tex]\( ab - dc \)[/tex]. Let’s rewrite the denominator correctly:
[tex]\[
\frac{1}{ad - bc} \left[\begin{array}{lr} -d & b \\ c & -a \end{array}\right]
\][/tex]
Comparing this with the standard formula:
[tex]\[
\frac{1}{ad - bc} \left[\begin{array}{cc} d & -b \\ -c & a \end{array}\right]
\][/tex]
We observe that the positions and signs inside the matrix given in option a do not match the standard positions and signs for the matrix inverse formula elements.
Therefore, based on the given structure, the correct formula for the inverse does not match option a, and hence option a is not the correct formula for finding the inverse of a 2x2 matrix.
