1. Matrices, Determinants, and Cramer’s Rule
Prepared by:
Dr. Fuad M.Shareef
Mahmoud Mahmoud
Cihan University - Erbil

2. 5.3 Determinant Solution of Linear Equations Determinants Cofactors
Evaluating n  n Determinants
Cramer’s Rule

3. Matrices Definition Example
A matrix is a rectangular array of numbers organized in rows and columns and encased in brackets.
Matrices are very useful due to the fact that they can be easily manipulated.
If A is an mxn martix -that is, a matrix with
m rows and n columns—then the scalar
entry in the ith row and jth column of A is
denoted by and is called the (i, j)-entry
of A.

4. Determinants
Every n  n matrix A is associated with a real number called the determinant of A, written  A. The determinant of a 2  2 matrix is defined as follows.

5. Determinant of a 2  2 Matrix
If A =

6. Note Matrices are enclosed with square brackets, while determinants
are denoted with vertical bars. A matrix is an array of numbers, but its determinant
is a single number.

7. Determinants
The arrows in the following diagram will remind you which products to find when evaluating a 2  2 determinant.

8. Solution EVALUATING A 2  2 DETERMINANT Example 1 Let A = Find A.
Use the definition with
a11
a22
a21
a12

9. Determinant of a 3  3 Matrix
If A =

10. Evaluating
The terms on the right side of the equation in the definition of A can be rearranged to get
Each quantity in parentheses represents the determinant of a 2  2 matrix that is the part of the matrix remaining when the row and column of the
multiplier are eliminated, as shown in the next slide.

11. Evaluating

12. Cofactors
The determinant of each 2  2 matrix above is called the minor of the associated element in the 3  3 matrix. The symbol represents Mij,the minor that results when row i and column j are eliminated. The following table in the next slide gives some of the minors from the previous matrix.

13. Cofactors
Element
Minor
a11
a22
a21
a23
a31
a33

14. Cofactors
In a 4  4 matrix, the minors are determinants of matrices. Similarly, an n  n matrix has minors that are determinants of matrices. To find the determinant of a 3  3 or larger matrix, first choose any row or column. Then the minor of each element in that row or column must be multiplied by +1 or – 1, depending on whether the sum of the row number and column number is even or odd. The product of a minor and the number +1 or – 1 is called a cofactor.

15. Cofactor Let Mij be the minor for element aij in an
n  n matrix. The cofactor of aij, written as Aij, is

16. a. 6 Solution FINDING COFACTORS OF ELEMENTS Example 2
Find the cofactor of each of the following elements of the matrix
a. 6
Solution
Since 6 is in the first row, first column of the matrix, i = 1 and j = 1 so
The cofactor is

17. b. 3 Solution FINDING COFACTORS OF ELEMENTS Example 2
Find the cofactor of each of the following elements of the matrix
b. 3
Solution
Here
i = 2 and j = 3 so,
The cofactor is

18. c. 8 Solution FINDING COFACTORS OF ELEMENTS Example 2
Find the cofactor of each of the following elements of the matrix
c. 8
Solution
We have,
i = 2 and j = 1 so,
The cofactor is

19. Finding the Determinant of a Matrix
Multiply each element in any row or column of the matrix by its cofactor.
The sum of these products gives the value of the determinant.

20. Use parentheses, & keep track of all negative signs to avoid errors.
EVALUATING A 3  3 DETERMINANT
Example 3
Evaluate
expanding by the second column.
Solution
Use parentheses, & keep track of all negative signs to avoid errors.

21. EVALUATING A 3  3 DETERMINANT
Example 3
Now find the cofactor of each element of these minors.

22. EVALUATING A 3  3 DETERMINANT
Example 3
Find the determinant by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products.

23. Cramer’s Rule
Determinants can be used to solve a linear system in the form
(1)
(2)
by elimination as follows.
Multiply (1) by b2 .
Multiply (2) by – b1 .
Add.

24. Cramer’s Rule Similarly, Multiply (1) by – a2 . Multiply (2) by a1 .
Add.

25. Cramer’s Rule
Both numerators and the common denominator of these values for x and y can be written as determinants, since

26. Cramer’s Rule
Using these determinants, the solutions for x and y become

27. Cramer’s Rule
We denote the three determinants in the solution as

28. replacing the coefficients of y in D by the respective constants.
Note The elements of D are the four coefficients of the variables in the
given system. The elements of Dx are obtained by replacing the coefficients
of x in D by the respective constants, and the elements of Dy are obtained by
replacing the coefficients of y in D by the respective constants.

29. Cramer’s Rule for Two Equations in Two Variables
Given the system
if then the system has the unique solution
where

30. Caution As indicated in the preceding box, Cramer’s rule does not apply if D = 0. When D = 0 the system is inconsistent or has infinitely many solutions. For this reason, evaluate D first.

31. Solution APPLYING CRAMER’S RULE TO A 2  2 SYSTEM Example 4
Use Cramer’s rule to solve the system
Solution
By Cramer’s rule, and Find D first, since if D = 0, Cramer’s rule does not apply. If
D ≠ 0, then find Dx and Dy.

32. APPLYING CRAMER’S RULE TO A 2  2 SYSTEM
Example 4
By Cramer’s rule,
The solution set is as can be verified by substituting in the given system.

33. General form of Cramer’s Rule
Let an n  n system have linear equations of the form
Define D as the determinant of the n  n matrix of all coefficients of the variables. Define Dx1 as the determinant obtained from D by replacing the entries in column 1 of D with the constants of the system. Define Dxi as the determinant obtained from D by replacing the entries in column i with the constants of the system. If D  0, the unique solution of the system is

34. Solution APPLYING CRAMER’S RULE TO A 3  3 SYSTEM Example 5
Use Cramer’s rule to solve the system.
Solution
Rewrite each equation in the form
ax + by + cz + = k.

35. APPLYING CRAMER’S RULE TO A 3  3 SYSTEM
Example 5
Verify that the required determinants are

36. APPLYING CRAMER’S RULE TO A 3  3 SYSTEM
Example 5
Thus,
and
so the solution set is

37. Caution As shown in Example 5, each equation in the system must be
written in the form ax + by + cz +    = k before using Cramer’s rule.

38. Solution SHOWING THAT CRAMER’S RULE DOES NOT APPLY Example 6
Show that Cramer’s rule does not apply to the following system.
Solution
We need to show that D = 0. Expanding about column 1 gives
Since D = 0, Cramer’s rule does not apply.

39. Note When D = 0, the system is either inconsistent or has infinitely many solutions. Use the elimination method to tell which is the case. Verify that the system in Example 6 is inconsistent, so the solution set is ø.