Does a singular n×n matrix always give infinitely many solutions?

[blinder]

I had a homework problem and i was wondering if anyone could help me out with it. It is a true or fasle question that requires explanation whether true or false. It goes like this...If A is a singular n by n matrix, then A*x=b has infinitely many solutions.(True or False)

 

[Data]

Is the matrix $N = (0)$ singular? Does $Nx = b$ have infinitely many solutions if b is nonzero?

 

[kleinwolf]

There are several case :

b=0, then Ax=0 has infinitely many solution if det(A)=0

b<>0 then it can have zero or an infinity of solution, depending on the determinants : $$det(b|_nA)$$

where $$b|_nA$$ means : the nth column of A is replace by b.

So the answer is "true and false" because you don't specify enough the question.

 

[mathwonk]

or you might say the question is meaningless since you do not quantify the letter "b".