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.

Expert Solution

milcah answered 2 years ago

Let A ∈ Mat n×n(R) be a real square matrix. (a) Suppose that A is symmetric,...

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Suppose C is a m × n matrix and A is a n × m matrix....

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Matrix A belongs to an n×n matrix over F. show that there exists a nonzero polynomial...

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Use this theorem to find the inverse of the given matrix or show that no inverse...

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A square matrix A is said to be symmetric if its transpose AT satisfies AT= A,...

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a. Let A be a square matrix with integer entries. Prove that if lambda is a...

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