[SOLVED] 5x5 real matrix

dwsmith

MHF Hall of Honor
Mar 2010
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Florida
Let M be a 5x5 real matrix. Exactly 4 of the following 5 conditions on M are equivalent to each other. Which one isn't?

(a) For any two distinct column vectors u and v of M, the set {u,v} is lin. ind.
(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 5x1 column vectors b
(d) The det(M) is nonzero
(e) There exists a 5x5 matrix N such that NM is the 5x5 identity matrix

b,c,d, and e are equivalent but why not a?
Since the matrix is nonsingular, the column vectors are lin ind.
 
Oct 2009
4,261
1,836
Let M be a 5x5 real matrix. Exactly 4 of the following 5 conditions on M are equivalent to each other. Which one isn't?

(a) For any two distinct column vectors u and v of M, the set {u,v} is lin. ind.
(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 5x1 column vectors b
(d) The det(M) is nonzero
(e) There exists a 5x5 matrix N such that NM is the 5x5 identity matrix

b,c,d, and e are equivalent but why not a?
Since the matrix is nonsingular, the column vectors are lin ind.

Of course, so (b,c,d,e)==>(a), but not the other way around! I bet you can find a singular matrix (i.e., determinant = 0) which fulfills this condition...

Tonio
 
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