Register

gate 2008 matrices engineering maths

Redirected
3,230 views
3 votes
3 votes
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
User Avatar

2 Answers

5 votes
5 votes

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.

User Avatar
1 votes
1 votes
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).
User Avatar
Voting & Accepting an answer helps improve the community
← PreviousNext →
← Previous in categoryNext in category →

Related questions

1 votes
1 votes
1 answer
1
jaswanth431 asked Dec 12, 2021
279 views
Self doubt: what are eigen values of resultant matrix when add two matrices?
if eigen values of matrix A = a,b,c if eigen values of matrix B = x,y,z then is it holds every time that eigen value of A+B = a+x, b+y, c+z ? please give some reference to you answer, thankyou.
if eigen values of matrix A = a,b,cif eigen values of matrix B = x,y,zthen is it holds every time that eigen value of A+B = a+x, b+y, c+z ?please give some reference to y...
User Avatar
0 votes
0 votes
0 answers
2
ankitgupta.1729 asked Jun 2, 2018
641 views
Gilbert Strang - Real Symmetric Matrices
Prove that :- For all real symmetric matrices , No. of positive Pivots = No . of positive eigenvalues Can anyone please give the formal mathematical proof for the above statement ?
Prove that :- For all real symmetric matrices , No. of positive Pivots = No . of positive eigenvaluesCan anyone please give the formal mathematical p...
User Avatar
0 votes
0 votes
1 answer
3
ankitgupta.1729 asked Nov 9, 2017
944 views
Matrices - LU Decompostion
Is LU Decomposition possible for every matrix ?
Is LU Decomposition possible for every matrix ?
User Avatar
5 votes
5 votes
2 answers
4
Shivi rao asked Oct 7, 2017
985 views
Matrices
Consider the following matrix $A=\begin{bmatrix} a&a^{2} &a^{3}-1 \\ b& b^{2} &b^{3}-1 \\ c& c^{2}&c^{3}-1 \end{bmatrix}$ if $|A|=0$, then the value of $abc$ is _______________
Consider the following matrix$A=\begin{bmatrix} a&a^{2} &a^{3}-1 \\ b& b^{2} &b^{3}-1 \\ c& c^{2}&c^{3}-1 \end{bmatrix}$if $|A|=0$, then the value of $abc$ is ___________...
User Avatar
Link Copied!