Elementary Matrices

Notice that inverses of elementary matrices are elementary matrices.

EA

Products of Elementary Matrices

A is a nonsingular matrix if and only if A is the product of elementary matrices of Type I, II, or III.

Column and Row Relationships

then linear relationships existing among columns of also hold among corresponding columns of B. That is,

Rank Normal Form

Equivalence

Testing for Equivalence

Corollary. Multiplication by nonsingular matrices cannot change rank

so, equivalence is only the measurement of rank

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