Page 17 - Vector Analysis

§1.4 Matrices 13

Definition 1.49. Let A P M(n, n; F) be a square matrix. A is said to be invertible if there
exists B P M(n, n; F) such that AB = In. The matrix B is called the inverse matrix of A,
and is usually denoted by A´1.

Proposition 1.50. Let A P M(n, n; F) be invertible. Then rank(A) = rank(A´1) = n.

Proof. Since A(A´1b) = (AA´1)b = b for all b P Fn, R(A) = Fn which implies that

rank(A) = n. We next show that R(A´1) = Fn. Denote A´1 by B, and let b P Fn.

Then BT(ATb) = (BTAT)b = b since BTAT = (AB)T = In. This observation implies that

R(BT) = Fn, and the theorem is then concluded by Theorem 1.48. ˝

Proposition 1.51. Let A P M(n, n; F) be invertible. Then A´1A = AA´1 = In.

Proof. We show that for all b P Fn, A´1Ab = b. Since A is invertible, rank(A´1) = n; thus
R(A´1) = Fn which implies that for each b P Fn, there exists x P F such that A´1x = b. As
a consequence,

(A´1A)b = (A´1A)(A´1x) = A´1(AA´1)x = A´1x = b . ˝

1.4.1 Elementary Row Operations and Elementary Matrices

Definition 1.52 (Elementary row operations). For an n ˆ m matrix A, three types of
elementary row operations can be performed on A:

1. The first type of row operation on A switches all matrix elements on the i-th row with
their counterparts on j-th row.

2. The second type of row operation on A multiplies all elements on the i-th row by a
non-zero scalar λ.

3. The third type of row operation on A adds j-th row multiplied by a scalar µ to the
i-th row.

The elementary row operation on an n ˆ m matrix A can be done by multiplying A by
an n ˆ n matrix, called an elementary matrix, on the left. The elementary matrices are
defined in the following

Definition 1.53 (Elementary matrices). An elementary matrix is a matrix which differs
from the identity matrix by one single elementary row operation.