Which one of the following statements is true?

Question

a) For all matrices A and B, if AB = I then BA = I
b) If A is a 3 \times 3 matrix, then det(2A) = 8 det(A).
c) For all matrices A and B, det(AB) = det(A) det(B).
d) If B is obtained by interchanging two rows of A, then det(B) = -1.

Answer 1

not (a) since in general, AB≠BA
not (b) since multiplying a row by n multiplies the determinant by n.
So, |nA| = n^2 |A|
not (d), since interchanging two rows makes |B| = -|A|

Better review the topic again.