Question

Let a be a 5 times 45×4 matrix. Explain why the equation ax​ = b cannot be consistent for all b in bold upper r superscript 5r5. Generalize your argument to the case of an arbitrary matrix a with more rows than columns.

Answer 1

A matrix with more rows than columns repesents a linear system with more equations than unknowns. Such a system is overdetermined, and it can either have no solutions (if some of the extra equations "contradict" each other), or it can have solution, for example if one equation is a multiple of another, and thus it has the same solution.

Allow me to use a 3x2 system, for the sake of brevity and graphical visualization. Let's start from a 2x2 system, with solution (1,1):

$\begin{cases} x+y=2\\x-y=0 \end{cases}$

Now we have to add the third equation. This equation makes the system overdetermined (more equations than unknown: 3 equations in 2 unknown), and we have two possible scenarios. If we add another equation that admits (1,1) as solution:

$\begin{cases} x+y=2\\x-y=0\\ 2x+3y=5 \end{cases}$

the system still admits a solution.

But if we add an equation which doesn't admit (1,1) as a solution:

$\begin{cases} x+y=2\\x-y=0\\ 2x+3y=18 \end{cases}$

The system becomes impossible.

Graphically, the first 3x2 system represents three lines all intersecting in the same point, which was the solution (1,1). The second 3x2 system represented three lines, where each couple of lines intersect in a different point. So, you can find a solution for each couple of lines, but you can find a global solution - a point belonging to all three lines.