College Algebra 9th txtbk

SECTION 7–5 Determinants and Cramer’s Rule 491
MATCHED PROBLEM 3
Evaluate
2 1 1
† 2 3 0†
1 2 1
It’s important to note that the determinant will work out the same regardless of which row
or column you choose to expand along. So if possible, you should choose a row or column
with one or more zeros to minimize the number of computations.
Z Using Cramer’s Rule to Solve Systems of Equations
Now we will see how determinants can be used to solve systems of equations. We’ll start
by investigating two equations in two variables, and then extend our results to three equations
in three variables.
Instead of thinking of each system of linear equations in two variables as a different
problem, let’s see what happens when we attempt to solve the general system
a 11 x a 12 y k 1
(3A)
a 21 x a 22 y k 2
(3B)
once and for all, in terms of the unspecified real constants a 11 , a 12 , a 21 , a 22 , k 1 , and k 2 .
We proceed by multiplying equations (3A) and (3B) by suitable constants so that when
the resulting equations are added, left side to left side and right side to right side, one of
the variables drops out. Suppose we choose to eliminate y. What constant should we use to
make the coefficients of y the same except for the signs? Multiply equation (3A) by a 22 and
(3B) by a 12 ; then add:
a 22 (3A):
a 12 (3B):
a 11 a 22 x a 12 a 22 y k 1 a 22
a 21 a 12 x a 12 a 22 y k 2 a 12
a 11 a 22 x a 21 a 12 x 0y k 1 a 22 k 2 a 12
(a 11 a 22 a 21 a 12 )x k 1 a 22 k 2 a 12
x k 1a 22 k 2 a 12
a 11 a 22 a 21 a 12
y is eliminated. Factor out x.
Solve for x.
a 11 a 22 a 21 a 12 0
At this point, the numerator and denominator might remind you of second-order determinants.
In fact, the value of x can be written as
x
Similarly, starting with system (3A) and (3B) and eliminating x (this is left as an exercise),
we obtain
y
k 1 a 12
` `
k 2 a 22
a 11 a 12
` `
a 21 a 22
a 11 k 1
` `
a 21 k 2
a 11 a 12
` `
a 21 a 22
These results are summarized in Theorem 2, Cramer’s rule, which is named after the
Swiss mathematician Gabriel Cramer (1704–1752).