Evaluate determinants of square matrices.
Use Cramer’s rule to solve systems of equations.
With every square matrix, we associate a number called its determinant.
Evaluate: |√2−3−4−√2|
=√2(−√2)−(−4)(−3)=−2−12=−14
Now Try Exercise 1.
We now consider a way to evaluate determinants of square matrices of order 3 × 3
Often we first find minors and cofactors of matrices in order to evaluate determinants.
For the matrix
find each of the following.
M11
M23
For M11
For M23
Now Try Exercise 9.
For the matrix given in Example 2, find each of the following.
A11
A23
In Example 2, we found that M11=−9
In Example 2, we found that M23=24
Now Try Exercise 11.
Consider the matrix A given by
The determinant of the matrix, denoted |A|
Because
we can write
It can be shown that we can determine |A|
Evaluate |A|
We have
The value of this determinant is −36
Now Try Exercise 13.
Determinants can be used to solve systems of linear equations. Consider a system of two linear equations:
Solving this system using the elimination method, we obtain
The numerators and the denominators of these expressions can be written as determinants:
If we let
we have
This procedure for solving systems of equations is known as Cramer’s rule.
Note that the denominator D contains the coefficients of x and y, in the same position as in the original equations. For x, the numerator is obtained by replacing the x-coefficients in D (the a’s) with the c’s. For y, the numerator is obtained by replacing the y-coefficients in D (the b’s) with the c’s.
Solve using Cramer’s rule:
We have
The solution is (−129,4129).
Now Try Exercise 29.
Cramer’s rule works only when a system of equations has a unique solution. This occurs when D≠0. If D=0 and Dx and Dy are also 0, then the equations are dependent. If D=0 and Dx and/or Dy is not 0, then the system is inconsistent.
Cramer’s rule can be extended to a system of n linear equations in n variables. We consider a 3×3 system.
Note that the determinant Dx is obtained from D by replacing the x-coefficients with d1, d2, and d3. Dy and Dz are obtained in a similar manner. As with a system of two equations, Cramer’s rule cannot be used if D=0. If D=0 and Dx, Dy, and Dz are 0, then the equations are dependent. If D=0 and one of Dx, Dy, or Dz is not 0, then the system is inconsistent.
Solve using Cramer’s rule:
We have
Then
The solution is (−2,35,125).
In practice, it is not necessary to evaluate Dz. When we have found values for x and y, we can substitute them into one of the equations to find z.
Now Try Exercise 37.
Evaluate the determinant.
1. |53−2−4|
2. |−86−12|
3. |4−7−23|
4. |−9−654|
5. |−2−√5−√53|
9. |√5−342|
7. |x4xx2|
8. |y2−2y3|
Use the following matrix for Exercises 9–16:
9.Find M11, M32, and M22.
10. Find M13, M31, and M23.
11.Find A11, A32, and A22.
12. Find A13, A31, and A23.
13. Evaluate |A| by expanding across the second row.
14. Evaluate |A| by expanding down the second column.
15.Evaluate |A| by expanding down the third column.
16. Evaluate |A| by expanding across the first row.
Use the following matrix for Exercises 17–22:
17.Find M12 and M44.
18. Find M41 and M33.
19.Find A22 and A34.
20. Find A24 and A43.
21.Evaluate |A| by expanding across the first row.
22. Evaluate |A| by expanding down the third column.
Evaluate the determinant.
23. |312−23134−6|
24. |3−21243−151|
25. |x0−12xx2−3x1|
26. |x1−1x2xx0x1|
Solve using Cramer’s rule.
27. −2x+4y=3,3x−7y=1
28. 5x−4y=−3,7x+2y=6
29. 2x−y=5,x−2y=1
30. 3x+4y=−2,5x−7y=1
31. 2x+9y=−2,4x−3y=3
32. 2x+3y=−1,3x+6y=−0.5
33. 2x+5y=7,3x−2y=1
34. 3x+2y=7,2x+3y=−2
35. 3x+2y−z=43x−2y+z=54x−5y−z=−1
36. 3x−y+2z=1,x−y+2z=3,−2x+3y+z=1
37. 3x+5y−z=−2,x−4y+2z=13,2x+4y+3z=1
38. 3x+2y+2z=1,5x−y−6z=3,2x+3y+3z=4
39. x−3y−7z=6,2x+3y+z=9,4x+y=7
40. x−2y−3z=4,3x−2z=8,2x+y+4z=13
41. 6y+6z=−1,8x+6z=−1,4x+9y=8
42. 3x+5y=2,2x−3z=7,4y+2z=−1
Determine whether the function is one-to-one, and if it is, find a formula for f−1(x). [5.1]
43. f(x)=3x+2
44. f(x)=x2−4
45. f(x)=|x|+3
46. f(x)=3√x+1
Simplify. Write answers in the form a+bi, where a and b are real numbers. [3.1]
47. (3−4i)−(−2−i)
48. (5+2i)+(1−4i)
49. (1−2i)(6+2i)
50. 3+i4−3i
Solve.
51. |y23y|=y
52. |x−3−1x|≥0
53. |2x112−134−2|=−6
54. |m+2−3m+5−4|=3m−5
Rewrite the expression using a determinant. Answers may vary.
55. a2+b2
56. 12h(a+b)
57. 2πr2+2πrh
58. x2y2−Q2