Question

# For each statement below, either show that the statement is true or give an example showing...

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.

(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.

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