De▯nition: Any matrix that can be obtained from an identity ma-
trix by means of one elementary row operation is called an elemen-
Theorem: Let A 2 M n;kR) and let E 2 M n;n(R) be elementary.
Multiplication of A on the left by E e▯ects the same elementary row
operation on A that was performed on the identity matrix to obtain
Note: If A s B, there are elementary matrices E ; E ;1▯▯▯2;E ‘
such that B = (E ▯▯‘E E )2. 1 De▯nition: An n▯n matrix A is invertible if there exists an n▯n
matrix B such that B A = AB = I. The matrix B is called the
inverse of A and is denoted by A . If A is not invertible, it is said
to be singular.
Note: We often write M nR) for M n;nR) when we are dealing with
Theorem: Let A 2 M (R).nIf B and C are matrices such that
AB = C A = I, then B = C.
In particular, if AB = B A = I, then B is the unique matrix with
Theorem: If A; B 2 M (R)nare invertible, then AB is invertible
▯1 ▯1 ▯1
and (AB) = B A . Fact: Every elementary matrix is invertible.
Lemma: Let A 2 M (n). The linear sy