Write the general form of a matrix equation in two unknowns.
See solution.
Practice makes perfect
Let's use the general form of a matrix equation in two unknowns x and y.
a& b c& d
x y = e f
Let's see what it means that there are infinitely many solutions.
This can only happen if the inverse of the coefficient matrix does not exist.
According to the formula of the inverse, this can only happen if the determinant is 0.
a& b c& d =ad-bc=0
Let's look for a rearrangement of this equation that can be translated to a verbal description.
We can write this last equation system in a matrix form.
a=kc b=kd
⟺
a& b =k c& d
Finally, let's interpret this result.
The matrix form means, that the first row of the coefficient matrix is a multiple of the second row.
To have a consistent system with infinitely many solutions, this property needs to extend to the constant matrix.
A system of two equations in two unknowns
has infinitely many solutions
if one equation is the multiple of the other.
Extra
Equation systems in more than two unknowns.
The following example shows, that for systems of equations in more than two unknowns the situation is more complicated.
Equations do not have to be multiples of each other for the system to have infinitely many solutions.
x+2y-3z=0 2x-y-z=0 3x+y-4z=0
You can check that any triple with x=y=z is a solution of this system.
Hence, there are infinitely many solution, but neither equation is a multiple of one of the others.
