Page 19 - Vector Analysis

§1.5 Determinants 15

3. Adding the j0-th row of A multiplied by a scalar µ to the i0-th row, where i0 ‰ j0, is
done by left multiplied A by the matrix E = [eij]nˆn given by

$ 1 if i = j,
&
eij = µ if (i, j) = (i0, j0),
% 0 otherwise,

or in the matrix form,


1 0 ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ 0

 0 1 0 ... µ 0  Ð the i0-th row
... ... ... ... 0
... ... 0
... ... 0 ... ... ...
... 0 ...
E = ... 1 0 ...
... ... ... 0

... ...
1

0 ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ 0 1

Ò

the j0-th column

Proposition 1.54. Every elementary matrix is invertible.

Theorem 1.55. Let A P M(n, n; F) be a square matrix. The following statements are
equivalent:

1. R(A) = Fn.

2. rank(A) = n.

3. Ax = b has a unique solution x for all b P Fn.

4. A is invertible.

5. A = EkEk´1 ¨ ¨ ¨ E2E1 for some elementary matrices E1, ¨ ¨ ¨ , Ek.

Proof. Note that by definition 1,2,3 are equivalent, and Proposition 1.50 shows that 4 ñ 2.

The implication from 3 to 4 is due to the fact that the map b ÑÞ x, where x is the unique

solution to Ax = b, is the inverse of A. Proposition 1.54 provides that 5 ñ 4. That 3 ñ 5

follows from that at most n(n + 1) elementary row operations has to be applied on A to

reach the identity matrix. ˝