Basis maths-linear algebra-Q17

+1 vote

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 May 15 in Basic Maths by gbeditor (44,490 points)
reshown May 16 by gbeditor

1 Answer

+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 11 by deepak-gatebook (112,390 points)