Question

For each statement below, either show that the statement is true or give an example showing that it is false. Assume throughout that A and B are square matrices, unless otherwise specified.

(a) If AB = 0 and A ̸= 0, then B = 0.

(b) If x is a vector of unknowns, b is a constant column vector, and Ax = b has no solution, then Ax = 0 has no solution.

(c) If x is a vector of unknowns and Ax = 0 has only the trivial solution, then, for every column vector b, the system Ax = b has a unique solution.

(d) If a system of simultaneous linear equations has more than one solution, then the reduced row echelon form of the system must have a row of zeroes.

(e) If in a system of simultaneous linear equations, there are more variables than equations, then there are infinitely many solutions.

Answer 1

(a). FALSE. If A =

1	0
0	0

and B =

0	0
1	0

then AB = 0, A≠0 and B≠0.

(b). FALSE. x = 0 i.e. the trivial solution is always a solution of the homogeneous equation Ax = 0.

( c). TRUE. If x = 0 is the only solution of the equation Ax = 0, then A has full rank so that A is invertible. Then x = A^-1 b is the unique solution to Ax = b regardless of the choice of b.

(d). TRUE. If the RREF of A does not have any zero row, then A has full rank so that A is invertible. Then x = A^-1 b is the unique solution to Ax = b. Therefore, if there is more than one solution, then the RREF of A must have a zero row.

(e). TRUE. In such a case, there will be some free variables so that there are infinitely many solutions.