Show that matrix A is not invertible by finding non trivial solutions

[black_hole]

Homework Statement

The 3x3 matrix A is given as the sum of two other 3x3 matrices B and C satisfying:1) all rows of B are the same vector u and 2) all columns of C are the same vector v.

Show that A is not invertible. One possible approach is to explain why there is a nonzero vector x satisfying both Bx = 0 and Cx = 0.

^^I'm having a hard time seeing why Bx=0 and Cx=0 should have nonzero solutions. I envision a matrix {{u1,u2,u3},{u1,u2,u3},{u1,u2,u3}} * some column vector = 0 but I'm just not seeing how to go about this when u1,u2,u3 could be anything.

Homework Equations

The Attempt at a Solution

 

[lmedin02]

Bx=0 and Cx=0 are homogeneous system of equations. So there are only two posibilities for their solutions. 1. Either the solutions are trivial and unique or 2. Infinitely many solutions.

Ask yourself if B is invertible? If it is, then multiply both sides of Bx=0 by its inverse to obtain only the trivial solution exists. If A is not invertible, then it must case 2.

Let me know if it helps.

 

[black_hole]

hmm, you've given me something to think about. So I can see how B is not invertible because well Gaussian elimination on it fails, but what about C? I don't think you can immediately see it by trying to do the more familiar elimination steps, so is it enough to say that in a way it a dependent matrix?

 

[vela]

Try writing out Cx=0 explicitly.

 

[lmedin02]

I am not sure what you mean by elemination failing. However, elimination give many rows of zero and hence infinitely many solutions for Bx=0. Consider doing elimination on C for a 2x2 matrix with the same columns and see what are your solutions.

If you have learn the determinant, then the determinant also gives you the answer imeddiately.

 

[Karnage1993]

First of all, what do you know about a matrix that has linearly dependent row or column vectors?

 

[black_hole]

Try writing out Cx=0 explicitly.

oh I see, if you try to eliminate one you end up eliminating all the others in the row as well

 

[black_hole]

I am not sure what you mean by elemination failing. However, elimination give many rows of zero and hence infinitely many solutions for Bx=0. Consider doing elimination on C for a 2x2 matrix with the same columns and see what are your solutions.

If you have learn the determinant, then the determinant also gives you the answer imeddiately.

Sorry, I should have been more clear. That is what I meant.

 

[black_hole]

Ok. I think I got it. Thank you all for your quick replies!