Page 175 - 35Linear Algebra

8.2 Elementary Matrices and Determinants 175

8.2.1 Row Swap

Our first elementary matrix swaps rows i and j when it is applied to a matrix
0
1
n
M. Explicitly, let R through R denote the rows of M, and let M be the
0
matrix M with rows i and j swapped. Then M and M can be regarded as
a block matrices (where the blocks are rows);
. . . . . .
   
i j
   
 R   R 
 .  0  . 
M =  .  and M =  .  .
 .   . 
R R
 j   i 
   
. . . . . .
Then notice that
     
1
.
.
 .   . . .   . 
 .     . 
j 0 1 i
     
 R     R 
0
M =  .  =  . . .   .  .

 . 
  . 
 .     . 
1 0
 i     j 
 R     R 
. .
   . .   
. . .
 .     . 
1
The matrix
 
1
.
 . . 
 
0 1
 
 
. i
 
 . .  =: E j
 
1 0
 
 
 . . 
 . 
1
i
is just the identity matrix with rows i and j swapped. The matrix E is an
j
elementary matrix and
0
i
M = E M .
j
Because det I = 1 and swapping a pair of rows changes the sign of the
determinant, we have found that
i
det E = −1 .
j
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