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

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

A is invertible.
The system Av=b has at least one solution for every column-vector b.
The system Av=b has exactly one solution for every column-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 identity matrix.
A is a product of elementary matrices.

Can you please help me to give some short examples or short proofs or logical arguments for EACh statements. EACH statements from 1-6 please.

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milcah answered 1 year ago

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