Page 18 - Vector Analysis

14 CHAPTER 1. Linear Algebra

1. Switching the i0-th and j0-th rows of A, where i0 ‰ j0, is done by left multiplied A
by the matrix E = [eij]nˆn given by

" 1 if (i, j) = (i0, j0) or (i, j) = (j0, i0) or i = j = k0 for some k0 ‰ i0, j0,
0 otherwise,
eij =

or in the matrix form,

 1 0 ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ 0
...
 0 ... 0 ... ... 
... ... 1 0 ...
... 0 ... ...
... 0 1 ... Ð the i0-th row
E = ... 1 1 ... 0 ... Ð the j0-th row
... 0 ... 1 ... ...
... ... 0 0 ...
0 ... 0
0 0 ...
1 ...
0

0 ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ 0 1

ÒÒ

the i0-th column the j0-th column

2. Multiplying the k0-th row of A by a non-zero scalar λ is done by left multiplied A by

the matrix E = [eij]nˆn given by

$ 0 if i ‰ j,
&
eij = λ if i = j = k0,
% 1 otherwise,

or in the matrix form,

1 0 ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ 0
 ... 
0 1 0 ... ...
... ... ... 1 ...
... 0 ...
... 0 0 ...
... λ ...
E = ... 0 0 0 Ð the k0-th row
...
1 0 ...
... ... 1

0

0 ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ ¨¨¨ 0 1