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134 Chapter 3 Matrix Algebra

Equivalence

• Whenever B can be derived from A by a combination of elementary
row and column operations, we write A ∼ B, and we say that A
and B are equivalent matrices. Since elementary row and column
operations are left-hand and right-hand multiplication by elementary
matrices, respectively, and in view of (3.9.3), we can say that
A ∼ B ⇐⇒ PAQ = B for nonsingular P and Q.

• Whenever B can be obtained from A by performing a sequence
row
of elementary row operations only, we write A ∼ B, and we say
that A and B are row equivalent. In other words,
row
A ∼ B ⇐⇒ PA = B for a nonsingular P.
• Whenever B can be obtained from A by performing a sequence of
col
column operations only, we write A ∼ B, and we say that A and
B are column equivalent. In other words,
col
A ∼ B ⇐⇒ AQ = B for a nonsingular Q.

If it’s possible to go from A to B by elementary row and column oper-
ations, then clearly it’s possible to start with B and get back to A because
elementary operations are reversible—i.e., PAQ = B =⇒ P −1 BQ −1 = A. It
therefore makes sense to talk about the equivalence of a pair of matrices without
regard to order. In other words, A ∼ B ⇐⇒ B ∼ A. Furthermore, it’s not
difficult to see that each type of equivalence is transitive in the sense that

A ∼ B and B ∼ C =⇒ A ∼ C.
In §2.2 it was stated that each matrix A possesses a unique reduced row
echelon form E A , and we accepted this fact because it is intuitively evident.
However, we are now in a position to understand a rigorous proof.
Example 3.9.2
Problem: Prove that E A is uniquely determined by A.
Solution: Without loss of generality, we may assume that A is square—
otherwise the appropriate number of zero rows or columns can be adjoined to A
row row
without affecting the results. Suppose that A ∼ E 1 and A ∼ E 2 , where E 1
row
and E 2 are both in reduced row echelon form. Consequently, E 1 ∼ E 2 , and
hence there is a nonsingular matrix P such that
PE 1 = E 2 . (3.9.4)