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I know what makes a system of equations have no solutions, but what leaves me confused are these matrices that are below. Do these matrices have an infinite number of solutions, since not every column have a pivot 1? Or is there a unique solution to both of these? $$ \left[ \begin{array}{ccc|c} 1 & 0 & 0 & a\\ 0 & 1 & 0 & b\\ 0 & 0 & 0 & 0 \end{array} \right] $$ or $$ \left[ \begin{array}{ccc|c} 0 & 0 & 0 & 0\\ 0 & 1 & 0 & b\\ 0 & 0 & 1 & c \end{array} \right] $$

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In the first case we can find $x=a$ and $y=b$ but we don't have conditions on $z$ which is free thus we have infinitely many solutions, that is the line $(a,b,0)+t(0,0,1)$.

And similarly for the second one for which the solutions is the line $(0,b,c)+t(1,0,0)$.

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