Basic Maths - Grand Test 1 -Q1

+2 votes

Let P be a matrix of order m \times n and Q be m \times 1 column vector (with real entries). Consider the equation Px = Q, x \in R^n admits a unique solution, then

(A) m=n

(B) m\leq n

(C) m\geq n

(D) m< n

asked Jun 21 in Basic Maths by gbeditor (21,290 points)
reshown Jun 23 by gbeditor

2 Answers

+1 vote
Best answer

Let P_{m\times n} has rank 'r' then there exists a unique solution in two cases:

(i) r = \text{no of unknowns (n) }= m \rightarrow (1)

(ii) r = \text{no of unknowns (n)} < m \rightarrow (2)

For example,

P_{m\times n} X_{n\times 1} = Q_{m\times 1}

(1) Let P_{4\times 4} X_{4\times 1} = Q_{4\times 1} here n = m

(2) Let P_{5\times 4} X_{4\times 1} = Q_{5\times 1} here n<m

Combining the above two equations

we get m\geq n


answered Jun 24 by gbeditor (21,290 points)
Px=Q is a non homogeneous equation and in that case
rank(P)=rank(P:Q)=r and
r=n then in that case we get unique solution
r<n then infinite solution
How these condition's is related to the given question. Please help me out sir.
i am also not getting the same
should not the second condition u mentioned above refers to infinite solution
0 votes
C is correct one
answered Jun 23 by tsnikhilsharmagate2018 (24,690 points)
how, can you explain?
Equation is always greater than equal to number of variable
So c is correct.