Question

Determine if each of the following statements is true or false. If it’s true, explain why. If it’s false explain why not, or simply give an example demonstrating why it’s false. (A correct choice of “T/F” with no explanation will not receive any credit.)

(a) If a system has more equations than variables, its RREF must have a row of 0’s.

(b) If a consistent system has more variables than equations, then it must have infinitely many solutions.

(c) Consider a system with m equations, n variables, and rank r. If the system has a unique solution, then m = n.

Answer 1

SOLUTION:

(a) If a system has more equations than variables, its RREF must have a row of 0’s.

Suppose a system has equations with variables where . Then its matrix is of size . We know that the rank of is less than or equal to and that is = (since ). In other words rank of is strictly less than .

The number of nonzero row in the RREF of = the number of linearly independent row of = Rank of the matrix < m

that means The RREF of must have a zero row. In particular RREF of must have exactly number of zero row.

Conclusion: The given statement is TRUE.

(b) If a consistent system has more variables than equations, then it must have infinitely many solutions.

Suppose a system has equations with variables where . If the system is consistent the we have . we also have . In other words (Since ).

We finally got that, the rank of is less than number of variables and system is consistent this together implies that the system has infinitely many solutions.

Conclusion: The given statement is TRUE.

(c) Consider a system with m equations, n variables, and rank r. If the system has a unique solution, then m = n.

Consider a linear system , where ,,

clearly we have (the number of variables). The system has unique solution that is . but in this example that is number of equation is not equals to number of variables.

Conclusion: The statement is FALSE.

REMARK:

Basic facts that are used in this above solution:

1. Let be matrix of size . Then rank of is less than minimum of . That is

2. Consider a linear system . where A be a matrix of size , and

a. The system is consistent if .Solution exist in this case.

• If , the system has UNIQUE solution.
• If , the system has INFINITELY MANY solution

b. The system is Inconsistent if . In this case system has no solution.