Page 74 - A Course in Linear Algebra with Applications

58 Chapter Three: Determinants

2 - 3
For example, |6| = 6 and 14.
4 1
Where does the expression a\\a 22 — a,\ 2a 2\ come from?
The motivation is provided by linear systems. Suppose that
we want to solve the linear system

anxi +012X2 = 61
a 2ixi + a 22x 2 = h

for unknowns x\ and x 2. Eliminate x 2 by subtracting a\ 2
times equation 2 from a 22 times equation 1; in this way we
obtain
- ai 2a 2i)xi = 61022 - a X2b 2.
(a na 22

This equation expresses xi as the quotient of a pair of 2 x 2
determinants:
b x a X2
b 2 a 22
a n aw
a 2i a 22

provided, of course, that the denominator does not vanish.
There is a similar expression for x 2.
The preceding calculation indicates that 2 x 2 determi-
nants are likely to be of significance for linear systems. And
this is confirmed if we try the same computation for a lin-
ear system of three equations in three unknowns. While the
resulting solutions are complicated, they do suggest the fol-
lowing definition for det(^4) where A = (0,^)3,3;

a
a n a 2 2 0 3 3 + <2l2<223 31 + Ql3«2ia32
a
—ai 2a 2id33 — ai3a 22a,3i — ana23 32