# Basic Maths - Grand Test 1 -Q1

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

$(A) m=n$

$(B) m\leq n$

$(C) m\geq n$

$(D) m< n$

reshown Jun 23

+1 vote

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

Combining the above two equations

we get $m\geq n$

answered Jun 24 by (4,380 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