Page 177 - 35Linear Algebra

8.2 Elementary Matrices and Determinants 177

Figure 8.3: Rescaling a row rescales the determinant.

Thus, multiplying a row by λ multiplies the determinant by λ. I.e.,

i
det R (λ)M = λ det M .

i
Since R (λ) is just the identity matrix with a single row multiplied by λ,
i
then by the above rule, the determinant of R (λ) is λ. Thus

1
 
.
 . . 
 
i  λ  = λ ,

det R (λ) = det 
.
 
 . . 
1
and once again we have a product of determinants formula

i
i

det R (λ)M = det R (λ) det M.
8.2.3 Row Addition

i
j
The final row operation is adding µR to R . This is done with the elementary
i
matrix S (µ), which is an identity matrix but with an additional µ in the i, j
j
position;
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