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"Let A be a 3 x 3 matrix with determinant 4. Then det(adj(${A^T)) = ?, det(adj(A^{-1}))}$ = ? and det(adj(4A)) = ?."

Are there any rules through which I can solve this? The fact that their are adjoints and determinants together is confusing me.

Any help will be highly appreciated!

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  • $\begingroup$ Do mean the adjoint or the adjugate matrix? $\endgroup$
    – amsmath
    Commented Oct 16, 2017 at 23:14
  • $\begingroup$ Adjoint. Sorry for any confusion. $\endgroup$
    – Dev SR
    Commented Oct 16, 2017 at 23:15
  • $\begingroup$ I am actually sure that you mean the adjugate. How do you define $adj(A)$? $\endgroup$
    – amsmath
    Commented Oct 16, 2017 at 23:17
  • $\begingroup$ Ok yes you are right! My apologies! I had never come across the term adjugate before so I was not sure! $\endgroup$
    – Dev SR
    Commented Oct 16, 2017 at 23:21
  • $\begingroup$ In your first comment you seemed to be very sure. You could at least have looked that up immediately. $\endgroup$
    – amsmath
    Commented Oct 16, 2017 at 23:23

1 Answer 1

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The adjugate matrix satisfies:

  1. $A^{-1}=\frac{1}{\det(A)} \text{adj}(A)$

Use (1) and the following properties of the determinant of a square matrix with dimensionality $n$:

  • $\det(A^{\top})=det(A)$
  • $\det(A^{-1})=\frac{1}{\det(A)}$.
  • $\det(cA)=c^n \det(A)$
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