To solve this problem, we'll use the properties of determinants specifically relating to the multiplication of matrices and the determinant of an inverse matrix.
If
A and
B are invertible matrices of the same order, then we know that:
The determinant of the product of two matrices is the product of their determinants, i.e.,
|AB|=|A||B|.
The determinant of the inverse of a matrix is the inverse of the determinant of the matrix, i.e.,
|(AB)−1|=‌Given that
|(AB)−1|=8 and
|A|=2, we need to find
|B|. Using the second bullet, we rearrange
|(AB)−1|=‌ to find
|AB| :
‌=8⇒|AB|=‌Now, using the first bullet point and the value of
|A|, we solve for
|B| :
‌|AB|=|A||B|‌‌=2⋅|B|‌|B|=‌÷2=‌Thus,
|B| is
‌, which corresponds to Option D.