This document discusses determinants and Cramer's rule. It defines determinants as special numbers associated with square matrices. It provides examples of calculating determinants of 2x2 and 3x3 matrices by rewriting columns and multiplying diagonals. The document also introduces Cramer's rule as a method to solve linear systems using determinants, where the variable is replaced with constants in the coefficient matrix. Examples are given to demonstrate solving 2x2 and 3x3 systems using Cramer's rule.

1. 4.3 DETERMINANTS AND
CRAMER’S RULE

2. WHAT IS A DETERMINANT?
A special number associated with a
square matrix.
Denoted “det A” or by |A|
Determinant of a 2 x 2 matrix:

3. 2 X 2 EXAMPLES
Evaluate the determinant of the matrix.

4. YOUR TURN!
Evaluate the determinant.

5. DETERMINANT OF A 3 X 3 MATRIX
Rewrite the first 2 columns to the right
of your matrix.
Multiply the diagonals.
Add/Subtract.

6. 3 X 3 EXAMPLE
Evaluate the determinant.

7. YOUR TURN!
Evaluate the determinant.

8. CRAMER’S RULE (2 X 2)
Used to solve linear systems.
Linear System:
Coefficient Matrix: = A
If det A 0,
Replace coefficients for the variable
you are finding with constants.

9. 2 X 2 EXAMPLES
Solve each system using Cramer’s rule.
8x + 5y = 2
2x – 4y = -10
4x – 6y = 4
x + 5y = 14

10. YOUR TURN!
Use Cramer’s rule to solve the system:
9x + 2y = 7
4x – 3y = 42

11. CRAMER’S RULE (3 X 3)
Let A be the coefficient matrix of the system:
If det A 0, then

12. 3 X 3 EXAMPLE
Solve the system using Cramer’s rule.
x + 3y – z = 1
-2x – 6y + z = -3
3x + 5y – 2z = 4