MML Discourse archived in May, 2026

Determinant of an inverse

mark

Let A be a non-singular matrix. Use the multiplicativity of the determinant to show that

\det(A^{-1}) = \frac{1}{\det(A)}.
User 009

Start with

A * A^{-1} = I

and

\det(A)*\det(A^{-1}) = \det(I)

divide both sides by

\det(A)

you get

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

and since

\det(I) = 1

the final equation is this:

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

That's great @User 025!

I did make one small edit to typeset your dets properly. Note that many mathematical functions have corresponding commands in LaTeX. Thus, typically we like to see

\det(A) \text{ rather than } det(A).

The first version is typed in as \det, rather than just det.