number of solutions of system of two equations, two unknowns (Matrix)

Question

How can we find that when a system of two equations, two unknowns has Infinite Solutions. I want a solution with matrix. I know this method (which is not with matrix):

$ax + by = c$

$a'x+ b'y = c'$

has infinite solutions if $\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}$$ (a,b,c,a',b',c' \neq 0)$ I want a solution just using matrix. I know if determinant is $0$ then the equation has infinite solutions or hasn't any solutions but how can we say it certainly has infinite solutions? Briefly, I want to know how can we determine a system of two equations two unknowns has infinite solutions with matrix concepts? Sorry for my English.

Answer 1

If $det(A) = 0$, then the columns are linearly dependent. Since there are only two columns, the columns must be scalar multiples of each other. There is a solution if and only if the right hand side vector is in the column space of the matrix, i.e. is a linear combination of the columns. In this case, that is true if and only if the right hand side vector is also a scalar multiple of the columns. This shows existence of at least one solution, and the determinant being zero implies there are infinitely many.

In summary: if $det(A) = 0$, then there are infinitely many solutions if and only if there exists a scalar $\alpha$ such that $\begin{bmatrix}c\\c'\end{bmatrix} = \alpha \begin{bmatrix}a\\a'\end{bmatrix}$. Otherwise there are no solutions.

Answer 2

You can use that a square system has unique solution iff $detA\neq0$, then one condition is that the $detA=0$. After that you have to find if there is any solution. To this you can use that the lines of the matrix are linear combination (in a $2\times2 $ matrix that means that the lines are in the form $L_1=kL_2$) and then check if the solution $c$ satisfies the same ratio.