(a)
Note that, det(AB) ≠ 0 means det(A) ≠ 0 and at the same time det(B) ≠ 0
Now det(B) ≠ 0 implies, B is invertible.
.
.
(b)
Since A is a n×n matrix and the equation Ax = 0 has a non-trivial solution then the row reduced echelon form of A has (n - 1) pivot columns.
This means that, Rank(A) = n - 1
Now as, n - 1 < n, so we have,
Rank(A) < n
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