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GR0568 36

Posted: Mon Oct 10, 2011 10:20 pm
by Hom
Let M be a 5 x 5 real matrix. Exactly four of the following five conditions on M are equivalent to each other. Which of the five conditions is equivalent to NONE of the other four?
(A) For any two distinct column vectors u and v of M, the set {u,v} is linearly independent.
(B) The homogeneous system Mx = 0 has only the trivial solution.
(C) The system of equations Mx = b has a unique solution for each real 5 x 1 column vector b.
(D) The determinant of M is nonzero.
(E) There exists a 5 x 5 real matrix N such that NM is the 5 x 5 identity matrix.

Answer:A

I don't have problem with B/C/D/E. But I don't know about A.
It should be right to say u,v are linearly independent. but what's the set{u,v} about?

Re: GR0568 36

Posted: Mon Oct 10, 2011 11:25 pm
by mjmiller2011
The last four answers are equivalent to saying that the matrix is invertible. The first one is not. Lets think of M as a 3x3 matrix, instead of a 5x5 for simplicity's sake. Let the left column read [0 1 1], the middle column read [0 0 1] and the right column read [0 1 0]. These column vectors are linearly independent, but the determinant of the matrix is zero.

Re: GR0568 36

Posted: Tue Oct 11, 2011 12:04 am
by blitzer6266
I think you meant this mjmiller2011, but it's important to mention that these column vectors are pairwise linearly independent, but not linearly independent as a set of 3 vectors.

Re: GR0568 36

Posted: Tue Oct 11, 2011 7:20 am
by Hom
mjmiller2011 wrote:The last four answers are equivalent to saying that the matrix is invertible. The first one is not. Lets think of M as a 3x3 matrix, instead of a 5x5 for simplicity's sake. Let the left column read [0 1 1], the middle column read [0 0 1] and the right column read [0 1 0]. These column vectors are linearly independent, but the determinant of the matrix is zero.
Thank you both. That's great example. And thank you for mentioning the key point.

Re: GR0568 36

Posted: Tue Oct 11, 2011 7:34 am
by mjmiller2011
Yeah that's what I meant to say, my bad for not being clear. In my example the set of three vectors is not linearly independent but any two vectors of the set are pairwise linearly independent.