Consider systems of m linear equations in n variables. I: If m n, none has a solution. III: If m = n, there exists a system with a solution. Which is correct?

GATE 2016 CS (CS2) – Question 14

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Consider systems of m linear equations in n variables. I: If m < n, all such systems have a solution. II: If m > n, none has a solution. III: If m = n, there exists a system with a solution. Which is correct?

I, II and III are true
Only II and III are true
Only III is true
None is true

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Correct answer: (C) Only III is true

Explanation

I fails because an underdetermined system can still be inconsistent, for example x+y=0 and x+y=1 with a third unused variable. II fails because extra equations may be redundant. III holds for the identity system Ix=b. Therefore only III is true, C.

GATE 2016 CS (CS2) – Question 14

Explanation

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