# Nontrivial solution for Ax=0 and Ax=b determine by pivot positions [closed]

A is a 3x2 matrix with two pivot positions.

(a) does the equation Ax=0 have a nontrivial solution

Since the two pivot positions will create 0 in the entire column in which they are present and 1 in its own position in reduced row echelon form and the rightmost column is all 0 therefore Ax=0 has no nontrivial solution

(b) does the equation Ax=b have atleast one solution for every possible b?

In the reduced row form b should have a [* * 0] form then only a unique non trivial solution exists

Is this correct and does it sound mathematical?

## closed as unclear what you're asking by Yves Daoust, Frits Veerman, Mathematician 42, Alex Provost, user223391 Apr 15 '17 at 15:12

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• It sounds hard to understand. What do you mean by "with two pivot positions" ? And if your matrix has three rows by two columns, your $[*\ *\ 0]$ is meaningless. – Yves Daoust Apr 12 '17 at 8:13
• @YvesDaoust, question is written as it is from the book and [* * 0] means a column vector with first and second non zero elements and last zero – Vikram Apr 12 '17 at 8:39
• I get what you are trying to say, but it is a bit jumbled. Please have a look at posting mathematical expressions to make your posts easier to follow. – hardmath Apr 12 '17 at 15:27

Your answer to (a) looks good. Question (b) can be asked alternately as $$Can $\mathbb{R}^3$ be spanned by only two vectors in $\mathbb{R}^3$$"$?