Page 176 - 35Linear Algebra

176 Determinants

Now we know that swapping a pair of rows flips the sign of the determi-
0
0
i
i
nant so det M = −detM. But det E = −1 and M = E M so
j
j
i
i
det E M = det E det M .
j
j
This result hints at a general rule for determinants of products of matrices.
8.2.2 Row Multiplication
The next row operation is multiplying a row by a scalar. Consider
 
R 1
.
M =  .  ,
 . 
R n
i
i
where R are row vectors. Let R (λ) be the identity matrix, with the ith
diagonal entry replaced by λ, not to be confused with the row vectors. I.e.,
 
1
.
 . . 
 
i  λ  .

R (λ) = 
.
 
 . . 
1
Then:
R
 1 
.
 . 
 . 
i
0
i
M = R (λ)M =  λR  ,


 . 
.
 . 
R n
equals M with one row multiplied by λ.
i
What effect does multiplication by the elementary matrix R (λ) have on
the determinant?
X
det M 0 = sgn(σ)m 1 σ(1) · · · λm i σ(i) · · · m n
σ(n)
σ
X 1 i n
= λ sgn(σ)m · · · m · · · m
σ(1) σ(i) σ(n)
σ
= λ det M

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