gate 2008 matrices engineering maths

Question

If the rank of a (5X6) matrix Q is 4, then which one of the following statement is correct

(A) Q  will have four linearly independent rows and four linearly independent columns  

(B) Q  will have four linearly independent rows and five linearly independent columns  

 (C) QQT will be invertible  

(D) QTQ will be invertible

Comments

rank is smallest sub determinant of a matrix whose value is non zero, other way of putting is linearly independent rows and columns.Looks like option A.

Answer 1

Given that rank Q is 4, so it will have 4 linearly independent columns as well as 4 linearly independent rows ($\because$ row rank of a matrix = column rank of a matrix)

So, A is correct and accordingly B is false.

C is false as order of $QQ^{T}$ will be 5 x 5 and given that rank of Q is 4 i.e. < 5. So the matrix is $QQ^{T}$ will be singular and hence not invertible.

Similarly D is false as order of $Q^{T}Q$ will be 6 x 6 and the matrix is $Q^{T}Q$ will be singular and hence not invertible.

Answer 2

Rank of a matrix is same as no of linearly independent row vectors as well as  no of linearly independent column  vectors in the matrix.

hence (A).