For row reduced echelon matrix for a homogeneous system of equations, what solution would be there for r=n and r>n

homogeneous equationlinear algebramatrices

I was learning Linear Algebra from Hoffman and Kunze. There the authors prove that for a row reduced echelon matrix with rows r and columns n for a homogeneous system of equations X, if r < n, X has a non-trivial solution, which I understood (substituting, r unknowns with remaining unknowns). However, what would happen in case of

  1. r = n
  2. r > n

For r = n, I am guessing it would be trivial solution only as no substitution is possible. Please correct me if wrong.

Best Answer

$Ax=0$ is homogeneous system.It always have trivial solution.If r=n then system will have trivial solution.As $rank(A)\le min{(r,n)}$. Hence r can not be greater than n.

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