2. Chapter 7: Matrices and Systems of Equations and Inequalities
7.2 Solution of Linear Systems in Three Variables
7.3 Solution of Linear Systems by Row Transformations
7.4 Matrix Properties and Operations
7.5 Determinants and Cramer’s Rule
7.6 Solution of Linear Systems by Matrix Inverses
7.7 Systems of Inequalities and Linear Programming
7.8 Partial Fractions

3. 7.5 Determinants and Cramer’s Rule
Subscript notation for the matrix A
The row 1, column 1 element is a11; the row 2, column 3 element is a23; and, in general, the row i, column j element is aij.

4. 7.5 Determinants of 2 × 2 Matrices
Associated with every square matrix is a real number called the determinant of A. In this text, we use det A.
The determinant of a 2 × 2 matrix A,
is defined as

5. 7.5 Determinants of 2 × 2 Matrices
Example Find det A if
Analytic Solution Graphing Calculator Solution

6. 7.5 Determinant of a 3 × 3 Matrix
The determinant of a 3 × 3 matrix A,
is defined as

7. 7.5 Determinant of a 3 × 3 Matrix
A method for calculating 3 × 3 determinants is found by re-arranging and factoring this formula.
Each of the quantities in parentheses represents the determinant of a 2 × 2 matrix that is part of the
3 × 3 matrix remaining when the row and column of the multiplier are eliminated.

8. 7.5 The Minor of an Element
The determinant of each 3 × 3 matrix is called a minor of the associated element.
The symbol Mij represents the minor when the ith row and jth column are eliminated.

9. 7.5 The Cofactor of an Element
To find the determinant of a 3 × 3 or larger square matrix:
Choose any row or column,
Multiply the minor of each element in that row or column by a +1 or –1, depending on whether the sum of i + j is even or odd,
Then, multiply each cofactor by its corresponding element in the matrix and find the sum of these products. This sum is the determinant of the matrix.
Let Mij be the minor for element aij in an n × n matrix. The cofactor of aij, written Aij, is

10. 7.5 Finding the Determinant
Example Evaluate det , expanding
by the second column.
Solution First find the minors of each element in the
second column.

11. 7.5 Finding the Determinant
Now, find the cofactor.
The determinant is found by multiplying each cofactor by its
corresponding element in the matrix and finding the sum of
these products.

12. 7.5 Cramer’s Rule for 2 × 2 Systems
Note: Cramer’s rule does not apply if D = 0.
For the system
where, if possible,

13. 7.5 Applying Cramer’s Rule to a System with Two Equations
Example Use Cramer’s rule to solve the system.
Analytic Solution By Cramer’s rule,

14. 7.5 Applying Cramer’s Rule to a System with Two Equations
The solution set is
Graphing Calculator Solution
Enter D, Dx, and Dy as matrices A, B, and C,
respectively.

15. 7.5 Cramer’s Rule for a System with Three Equations
For the system
where