(a) is false. Consider the identity matrix, . Then
bu
(b) is false. Since is invertible, we have
but in general, matrix multiplication is non-commutative.
(c) is true. Let ; then
and so on, down to
which is to say .
(d) is false. Consider , so that
which is singular (i.e. determinant is zero), and thus not invertible.
(e) is true. It follows from the distributivity of matrix multiplication:
(f) is false for the same reason as (b). Expanding the product gives
but in general, .