How Do You Calculate the Determinant of Complex Matrix Expressions?

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To calculate the determinant of complex matrix expressions involving a 3x3 matrix A with det(A)=15, one can apply properties of determinants and inverses. The expression det[A^3((adj(A))−1)²] can be simplified using the determinant of the adjugate and the inverse of A. Similarly, for det[5A^−1(adj(A))], the determinant properties allow for simplification. The key challenge lies in correctly applying these properties to arrive at numerical values. Understanding these determinant properties is crucial for solving such matrix problems effectively.
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Homework Statement



Suppose A is a 33 matrix such that det(A)=15.

Then det[A3((adj(A))−1)2]= and det[5A−1(adj(A))]

-1=inverse

Homework Equations



I know the properties of determinants and inverses

The Attempt at a Solution



Problem simplifying to get a number.
 
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If you know the properties of determinants and inverses, can't you at least explain what the problem you are having is?
 

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