Page 178 - 35Linear Algebra

178 Determinants

 
1
.
 . . 
 
1 µ
 
 
i
S (µ) =  . . .  .


j
 
1
 
 
 . . 
 . 
1
i
Then multiplying M by S (µ) performs a row addition;
j
     
1
. .
.
 . .   .   . . 
   .   
1 µ
   i   i j 
   R   R + µR 
.
   .   . 
 . .   .  =  . .  .
   .   
1 R
   j   j 
   R   
 . .   .   . . 
.
 .   .   . 
1
0
i
What is the effect of multiplying by S (µ) on the determinant? Let M =
j
00
i
j
i
S (µ)M, and let M be the matrix M but with R replaced by R Then
j
X j
det M 0 = sgn(σ)m 1 σ(1) · · · (m i σ(i) + µm σ(i) ) · · · m n
σ(n)
σ
X
= sgn(σ)m 1 · · · m i · · · m n
σ(1) σ(i) σ(n)
σ
X j j
+ sgn(σ)m 1 σ(1) · · · µm σ(j) · · · m σ(j) · · · m n
σ(n)
σ
= det M + µ det M 00
00
Since M has two identical rows, its determinant is 0 so
0
det M = det M,
0
when M is obtained from M by adding µ times row j to row i.
Reading homework: problem 3
178