(Linear Algebra) Suppose a is a 3x3 matrix, with det(a) = 3?

Question

compute det(A^(-1)).

Is the answer 1/3? I m still confused how to solve this?

Answer 1

Remember the property: det(AB) = det(A) det(B)

Then let B = A^-1, so that AB = I. Then

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

...and that's equal to 1/3 in this case where det(A) = 3.

Answer 2

Yes, the determinate of the inverse of a matrix is the reciprocal of the determinate of the matrix.