Page 398 - 35Linear Algebra

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• Scalar multiplying:

λ 0 1 λ 0 a b λa λb
1
R (λ) = , E M = = .
2
0 1 0 1 c d c d
• Row sum:

1 λ 1 1 λ a b a + λc b + λd
1
S (λ) = , S (λ)M = = .
2 0 1 2 0 1 c d c d
Elementary Determinants
This video will show you how to calculate determinants of elementary matrices.
First remember that the job of an elementary row matrix is to perform row
operations, so that if E is an elementary row matrix and M some given matrix,

EM

is the matrix M with a row operation performed on it.
The next thing to remember is that the determinant of the identity is 1.
Moreover, we also know what row operations do to determinants:
i
• Row swap E : flips the sign of the determinant.
j
i
• Scalar multiplication R (λ): multiplying a row by λ multiplies the de-
terminant by λ.
i
• Row addition S (λ): adding some amount of one row to another does not
j
change the determinant.
The corresponding elementary matrices are obtained by performing exactly
these operations on the identity:
 
1
.
 
.
.
 
 
0 1
 
 
i  . 
E =   ,
j .
 . 
 
1 0
 
 
.
 
.
.
 
1
1
 
.
 
.
. 
 
i

R (λ) =  λ  ,
 
 . 
.
 
.
1
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