Linear Algebra, Theory And Applications, 2012a

3.1. BASIC TECHNIQUES AND PROPERTIES 79
where the i th row of A 1 is (a 1 , ··· ,a n ) and the i th row of A 2 is (b 1 , ··· ,b n ) , all other rows
of A 1 and A 2 coinciding with those of A. In other words, det is a linear function of each
row A. The same is true with the word “row” replaced with the word “column”. In addition
to this, if A and B are n × n matrices, then
and if A is an n × n matrix, then
det (AB) =det(A)det(B) ,
det (A) =det ( A T ) .
This theorem implies the following corollary which gives a way to find determinants. As
I pointed out above, the method of Laplace expansion will not be practical for any matrix
of large size.
Corollary 3.1.9 Let A be an n×n matrix and let B be the matrix obtained by replacing the
i th row (column) of A with the sum of the i th row (column) added to a multiple of another
row (column). Then det (A) =det(B) . If B is the matrix obtained from A be replacing the
i th row (column) of A by a times the i th row (column) then a det (A) =det(B) .
Here is an example which shows how to use this corollary to find a determinant.
Example 3.1.10 Find the determinant of the matrix
⎛
⎞
1 2 3 4
A = ⎜ 5 1 2 3
⎟
⎝ 4 5 4 3 ⎠
2 2 −4 5
Replace the second row by (−5) times the first row added to it. Then replace the third
row by (−4) times the first row added to it. Finally, replace the fourth row by (−2) times
the first row added to it. This yields the matrix
B =
⎛
⎜
⎝
1 2 3 4
0 −9 −13 −17
0 −3 −8 −13
0 −2 −10 −3
and from the above corollary, it has the same determinant as A. Now using the corollary
some more, det (B) = ( )
−1
3 det (C) where
⎛
⎞
1 2 3 4
C = ⎜ 0 0 11 22
⎟
⎝ 0 −3 −8 −13 ⎠ .
0 6 30 9
The second row was replaced by (−3) times the third row added to the second row and then
thelastrowwasmultipliedby(−3) . Now replace the last row with 2 times the third added
to it and then switch the third and second rows. Then det (C) =− det (D) where
D =
⎛
⎜
⎝
1 2 3 4
0 −3 −8 −13
0 0 11 22
0 0 14 −17
⎞
⎟
⎠
⎞
⎟
⎠