GR0568 36
Posted: Mon Oct 10, 2011 10:20 pm
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?
(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?