Matrices and Determinants

Given that |adj‌A|=x and |adj‌B|=y, we are tasked with finding |(adj(AB))−1|.
First, recognize that the determinant of the adjugate of a matrix can be expressed with respect to the determinants of the matrix itself. Using this relation, we have:
|adj(AB)|=|adj(adj(A))|⋅|adj(adj(B))|
The inverse of the adjugate adjoint product can be decomposed as:
|(adj(AB))−1|=∣(adj‌B)−1(adj‌A)−1)∣
Because |adj‌A|=x and |adj‌B|=y, the properties of determinants give us:
‌|(adj‌A)−1|=‌

1

|adj‌A|

=‌

1

x

‌|(adj‌B)−1|=‌

1

|adj‌B|

=‌

1

y

Thus, the determinant of the inverse of the product is:
|(adj(AB))−1|=‌

1

x

×‌

1

y

=‌

1

xy

Therefore, the result is ‌

1

xy

.