Theorem. If A is an n by n squarematrix, then the following statements are equivalent.
- A is invertible.
- The system Av=b has at least one solution for everycolumn-vector b.
- The system Av=b has exactly one solution for everycolumn-vector b
- The system Av=b has only the trivial solution(0,0,0,...0) ( Homogeneous systems).
- The reduced row-echelon form of A is the identitymatrix.
- A is a product of elementary matrices.
Can you please help me to give some short examples or shortproofs or logical arguments for EACh statements. EACH statementsfrom 1-6 please.
