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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?

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

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ 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. $\endgroup$ – Yves Daoust Apr 12 '17 at 8:13
  • $\begingroup$ @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 $\endgroup$ – Vikram Apr 12 '17 at 8:39
  • $\begingroup$ 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. $\endgroup$ – hardmath Apr 12 '17 at 15:27
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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$$"$?

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