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Sunday, November 19, 2017

Engineering Mathematics, Linear Algebra, 2016(1)



1. Consider the systems, each consisting of m linear equations in n variables.
  1. if m < n, then all such systems have a solutions.
  2. If m>n, then non of these systems has a solutions. 
  3. if m=n, then there exits a system which has a solution.
Which one of the following is CORRECT?
  1. i,ii and iii are true
  2. Only ii and iii are true
  3. Only iii is true
  4. None of them is true.

Solution:

Statement i
Consider 2 equations in three variables.
i.e., m=2 & n=3 (m<n)
x - y + z = 1
-x + y -z = 2

This system has no solutions is inconsistent (x=1 and y=1). 
i.e., "For ‘n’ variables, we need atleast ‘n’ linear equations  to find value of each variable"

∴ Statement i is false.


Statement ii
Consider 3 equations in two variables.
i.e., m=3 & n=2 (m>n)

x + y = 2,
x - y = 0,
3x + y = 4

This system has a unique solutions (no solution)
unique solutions : "A system of linear equations ax + by + c = 0 and dx + ey + g = 0 will have a unique solution if the two lines represented by the equations ax + by + c = 0 and dx + ey + g = 0 intersect at a point. i.e., if the two lines are neither parallel nor coincident. Essentially, the slopes of the two lines should be different."

∴ Statement ii is false.

Statement iii.
Consider a system with 2 equations and two variables.
i.e, m=2 & n=2 (m=n)

x + y = 2 and x - y =0
The system has a solution x = 1 and y = 1 

∴ Statement iii is true.

Hence ANSWER is c


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