Given an invertible square matrix #A#, what is #det(A^(-1))# in terms of #det(A)#?

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Answer 1 · 2017-12-25T22:47:31

#det(A^(-1)) = 1/det(A)#

If #A# is any invertible matrix, then:

#det(A^(-1)) = 1/det(A)#

The determinant has a multiplicative property:

#det(AB) = det(A) * det(B)#

Also:

#det(I) = 1#

So we find:

#det(A) * det(A^(-1)) = det(A A^(-1)) = det(I) = 1#

Hence:

#det(A^(-1)) = 1/det(A)#