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Theorem. If A is an n by n square matrix, then the following statements are equivalent. A...

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Basic Math

Theorem. If A is an n by n squarematrix, then the following statements are equivalent.

  1. A is invertible.
  2. The system Av=b has at least one solution for everycolumn-vector b.
  3. The system Av=b has exactly one solution for everycolumn-vector b
  4. The system Av=b has only the trivial solution(0,0,0,...0) ( Homogeneous systems).
  5. The reduced row-echelon form of A is the identitymatrix.
  6. 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.

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