Suppose that A, B and C are m n matrices. Use the definition of row-equivalence

Question

Suppose that A, B and C are m n matrices. Use the definition of row-equivalence
to prove the following three facts:

1. A is row-equivalent to A.
2. If A is row-equivalent to B, then B is row-equivalent to A.
3. If A is row-equivalent to B, and B is row-equivalent to C, then A is row-equivalent to C.

Hint: A relationship that satisfies these three properties is known as an equivalence relation, an important idea in the study of various algebras. This is a formal way of saying that a relationship behaves like equality, without requiring the relationship to be as strict as equality itself.

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Explanation / Answer

suppose A, B, & C are m x n matrices.

row equivalence:  two matrices are row equivalent if a series of elementary row operations can transform one matrix to another

this series of elementary row operations can be represented by the matrices Ei

so, if matrix A is row equivalent to matrix B, then we could write

E1E2...EnA = B

(1) A is row equivalent to A

proof:

the identity row operation I can simply be applied to A to obtain A

i.e. IA = A

thus, A is row equivalent to A

(2) if A is row equivalent to B then B is row equivalent to A

proof:

Assume A is row equivalent to B.

Then E1E2...EnA = B

since the inverse of an elementary matrix E is itself elementary, we can write

A = En-1...E2-1E1-1B

thus, a series of elementary row operations transforms B to A, and B is row equivalent to A

(3)  If A is row-equivalent to B, and B is row-equivalent to C, then A is row-equivalent to C.

proof:

assume A is row-equivalent to B and B is row-equivalent to C.

then

E1E2...EnA = B

D1D2...DnB = C

and we see that

D1D2...DnE1E2...EnA = C

and that A is row equivalent to C.

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