In this determinant, and Multiply each of these num-
bers by its minor, and combine the three terms using the definition. Notice that the
second term in the definition is subtracted.
=- 6 NOW TRY
=- 1 + 8 - 13
= 11 - 12 + 1182 + 11 - 132
+ 1331 - 32 - 1 - 221 - 224
= 13 - 2122 - 1 - 3214 + 133122 - 1 - 2214
3
1
- 1
1
3
- 2
1
- 2
- 3
2
(^3) = 1 `-^2
1
- 3
2
- 1 - 12
3
1
- 2
2
+ 1
3
- 2
- 2
- 3
`
a 1 =1,a 2 =-1, a 3 =1.
APPENDIX A Determinants and Cramer’s Rule 717
Use parentheses
and brackets to
avoid errors.To obtain equation (1), we could have rearranged terms in the definition of the
determinant and used the distributive property to factor out the three elements of the
second or third column or of any of the three rows. Expanding by minors about any
row or any column results in the same value for a determinant.
To determine the correct signs for the terms of other expansions, the array of
signsin the margin is helpful. The signs alternate for each row and column beginning
with a in the first row, first column position. For example, if the expansion is to be
about the second column, the first term would have a minus sign associated with it,
the second term a plus sign, and the third term a minus sign.
Evaluating a 3 3 DeterminantEvaluate the determinant of Example 2using expansion by minors about the second
column.
The result is the same as in Example 2.
NOW TRYOBJECTIVE 3 Understand the derivation of Cramer’s rule.We can use
determinants to solve a system of equations of the form
(1)
(2)The result will be a formula that can be used to solve any system of two equations
with two variables.
Multiply equation by
Multiply equation by
Add.x = 1 if a 1 b 2 - a 2 b 1 Z 02
c 1 b 2 - c 2 b 1
a 1 b 2 - a 2 b 1
1 a 1 b 2 - a 2 b 12 x= c 1 b 2 - c 2 b 1
- a 2 b 1 x-b 1 b 2 y=-c 2 b 1 122 - b 1.
a 1 b 2 x+b 1 b 2 y= c 1 b 2 112 b 2.
a 2 x+ b 2 y= c 2.
a 1 x+ b 1 y= c 1
=- 6
=- 3 - 8 + 5
=- 3112 - 2142 - 11 - 52
3
1
- 1
1
3
- 2
1
- 2
- 3
2
3 = - 3 `
- 1
1
- 3
2
+ 1 - 22
1
1
- 2
2
- 1
1
- 1
- 2
- 3
`
EXAMPLE 3 :
+
3 : 3
Solve for x.Array of Signs for a
3 3 Determinant:
NOW TRY
EXERCISE 2
Evaluate the determinant by
expansion by minors about
the first column.
(^3)
0 - 23
41 - 5
6 - 15
3
NOW TRY ANSWERS
- 70 3. 70
NOW TRY
EXERCISE 3
Evaluate the determinant by
expansion by minors about
the second column.
(^3)