Determining whether the system will have a nontrivial solution?

Question

Say I have a 3x3 matrix (a1 = 3a2 - 2a3), Will they system Ax=b have a nontrivial solution? Is it non-singular?

I realize nontrivial means an answer that is not a zero vector.

It must be the variables that are confusing me. I know that singular matrices need to have a 0 determinant, but not sure how to figure it out with this information. Thanks.

Answer 1

If I understand the notation correctly, $a_1=3a_2-2a_3$ means the rows of $A_{3 \times 3}$ are linearly dependent; this implies the matrix is singular.

Whether or not $Ax=b$ has a non-trivial solution depends on $b$.

• If $b$ belongs to the column space of $A$, then there will be infinitely many (non-trivial) solutions. By definition, there will be one solution, and $x+y$ will be another for all $y$ in the null space of $A$ (which is non-trivial, since $A$ is singular).

• Otherwise $b$ does not belong to the column space of $A$, so, by definition, there will be no solutions.

Answer 2

Please give Rebecca credit by voting her answer up. It is just courtesy.

To expand on her answer, the system of equation will have a nontrivial soltio if and only if $$ b_1 = 3b_2-2b_3$$ I.e. $b$ has the same linear dependence of rows.