One simple question about determinant..

[kochibacha]

consider non-homogenous system

ax+by = k₁
cx+dy = k₂
if determinant is zero then it can have infinitely many solution or no solution at all

my question is WHEN THERE IS NO SOLUTION TO THIS SYSTEM?

of course , there must be something related to k1 and k2

 

[Deleted member 4993]

consider non-homogenous system

ax+by = k₁
cx+dy = k₂
if determinant is zero then it can have infinitely many solution or no solution at all

my question is WHEN THERE IS NO SOLUTION TO THIS SYSTEM?

of course , there must be something related to k1 and k2

The system has no solution, if:

a = n * b and b = n*d and k₁ \(\displaystyle \ne\) n*k₂

Note that a = n * b and b = n*d → determinant = 0

 

[HallsofIvy]

If n by n matrix A has determinant 0, then it maps all of \(\displaystyle R^n\) into a subspace of \(\displaystyle R^n\) (of dimension m where m is the "rank" of matrix A). In that case the equation Ax= b has a solution (in fact an infinite number of solutions) if and only if b lies in that subspace.