Question

Let A be a nonsingular matrix. Show that det(A^-1) = 1/det(A)

Answer 1

\[\det(A ^{-1})=\frac{ 1 }{ \det(A) }\]

I'm assuming that it would have something to do with using the elementary matrices and how \[E_{k}...E_{1}A=I_{n}\] where \[E_{k}...E_{1}=A^{-1}\] but beyond that not sure how to set it up

Answer 2

Determinant of a product of two matrices is the product of their determinants :D

Answer 3

Yes. But, how would I show that this is true in general for all nonsingular matrices? I could set the determinant of each side equal to each other, in which case assuming that the claim is true, but still have no idea how I would show that it is actually true

Answer 4

But \[\Large \det[AB]=\det[A]\det[B]\]?

Answer 5

Yes. and I am told to PROVE that \[det(A^{-1}) = \frac{1} {det{A}}\]
Wouldn't just saying that \[det(E_{k})...det(E_{1})det(A)=det(I_{n})\] be doing nothing more than assuming the initial statement as true and not prove anything?

Answer 6

\[\Large A^{-1}A = I_n\]

Answer 7

\[\Large \det[A^{-1}A]=\det[I_n]\]...

Answer 8

\[\Large \det[I_n]=1\]\[\Large \det[A^{-1}A]= \det[A^{-1}]\det[A]\]
stop me whenever ^.^

Answer 9

okay. Sometimes I'm not quite sure how I've come this far in math...

Answer 10

^.^