Question

In what follows, A and B denote 2 x 2 matrices. Answer each question below, with justification. No one answer should be more than a few lines long.

A1. If k is a scalar, how does the determinant of kA relate to the determinant of A?

A2. Show that the determinant of A + B is not necessarily the same as det A + det B. (Remark: a single specific counterexample suffices!)

A3. If A is singular and B is any matrix whatsoever, is AB singular or non-singular?

A4. If B is obtained from A by interchanging two rows of A, how does det A relate to det B?

A5. If det A = -5, what is det A^-1?

Answer 1