Page 271 - Advanced engineering mathematics

8.1 Definition of the Determinant 251

⎛ ⎞ ⎛ ⎞
a 11 a 12 ··· a 1n a 11 a 12 ··· a 1n
⎜ ··· ··· ··· ··· ⎟ ⎜ ··· ··· ··· ··· ⎟
⎜ ⎟ ⎜ ⎟
a i1 a i2 a in a i1 a i2 a in
⎜ ··· ⎟ ⎜ ··· ⎟
⎜ ⎟ ⎜ ⎟
= ⎜ ··· ··· ··· ··· ⎟ + ··· ··· ··· ··· .
⎟
⎜
⎜ ⎟ ⎜ ⎟
αa i1 αa i2 ··· αa in a k1 a k2 ··· a kn
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎝ ··· ··· ··· ··· ⎠ ⎝ ··· ··· ··· ··· ⎠
a n1 a n2 ··· a in a n1 a n2 ··· a in
Then |A| is the sum of the determinants of the matrices on the right. But the second determinant
on the right is just |A| and the first is 0 by (4) because row k is a multiple of row i.
For conclusion (9), note that, by (3), (5) and (8), every time we produce B from A by an
elementary row operation, |B| is equal to a nonzero multiple of A. Since we reduce a matrix by
a sequence of elementary row operations, then |A| is always a nonzero multiple of |A R |.This
means that |A| is nonzero if and only if |A R | is nonzero. But this is the case exactly when A is
nonsingular, since in this case A R = I n .If A R = I n , then A R has at least one zero row and has
determinant zero.
Vanishing or non-vanishing of the determinant is an important test for existence of an
inverse, and we will use it when we discuss eigenvalues in the next chapter.
Finally, we will sketch a proof of conclusion (10). If A is nonsingular, then there is a product
of elementary matrices that reduces A to I n :
E r E r−1 ···E 1 A = I n .

Then
−1
−1
−1
A = E E ···E ,
1 2 r
a product of inverses of elementary matrices, which are again elementary matrices. Since we
can do this for nonsingular B as well, we can write AB as a product of elementary matrices. It
is therefore sufficient to show that the determinant of a product of elementary matrices is the
product of the determinants of these elementary matrices. This can be done for two elementary
matrices using properties (3), (5) and (8) of determinants then extended to arbitrary products by
induction.
If either A or B is singular, then so is AB, and in this case,
|AB|= 0 =|A||B|.

Conclusions (3), (5), and (8) tell us the effects of elementary row operations on the deter-
minant of a matrix. However, in the context of determinants, these operations can be applied
to columns as well. When we use matrices to represent systems of equations, rows contain
equations and columns contain coefficients of particular unknowns, so there is an essential dif-
ference between rows and columns. However, the determinant of a matrix does not involve these
interpretations and there is no preference of rows over columns (for example, |A|=|A |).
t

SECTION 8.1 PROBLEMS

1. Let A =[a ij ] be an n × n matrix and let α be a number. and |B| related? Hint: It is useful to look at the 2 × 2
Form B =[αa ij ] by multiplying each element of A by and 3 × 3 cases to get some idea of what B looks like.
α.How are |A| and |B| related? t
3. An n ×n matrix is skew-symmetric if A=−A . Explain
2. Let A =[a ij ] be an n × n matrix. Let α be a nonzero why the determinant of a skew symmetric matrix hav-
number. Form the matrix B =[α i− j a ij ].How are |A| ing an odd number of rows and columns must be zero.

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October 14, 2010 14:26 THM/NEIL Page-251 27410_08_ch08_p247-266