1. 3.7 – Evaluate Determinants and Apply Cramer’s Rule
Associated with each square (n x n) matrix is a real number called its determinant. The determinant of a matrix A is denoted by det A or by |A|.
The determinant of a 2 x 2 matrix is the difference of the products of the elements of the diagonals.

2. 3.7 – Evaluate Determinants and Apply Cramer’s Rule

3. 3.7 – Evaluate Determinants and Apply Cramer’s Rule
Example 1:
Evaluate the determinant of the matrix.

4. 3.7 – Evaluate Determinants and Apply Cramer’s Rule
Example 2:
Evaluate the determinant of the matrix.

5. 3.7 – Evaluate Determinants and Apply Cramer’s Rule
You can use a determinant to find the area of a triangle whose vertices are points in a coordinate plane.

6. 3.7 – Evaluate Determinants and Apply Cramer’s Rule
Example 3:
Off the coast of California lies a triangular region of the Pacific Ocean where huge populations of sea lions and seals live. The triangle is formed by imaginary lines connecting Bodega Bay, the Farallon Islands, and Ano Nuevo Island, as shown (next slide). Use the determinant to estimate the area of the region.

7. 3.7 – Evaluate Determinants and Apply Cramer’s Rule
Example 3:
Units measured in miles.

8. 3.7 – Evaluate Determinants and Apply Cramer’s Rule
You can use determinants to solve a system of linear equations. The method, called called Cramer’s rule and named after the Swiss mathematician Gabriel Cramer, uses the coefficient matrix of the linear system.

9. 3.7 – Evaluate Determinants and Apply Cramer’s Rule

10. 3.7 – Evaluate Determinants and Apply Cramer’s Rule
Example 4:
Use Cramer’s rule to solve the system:

11. 3.7 – Evaluate Determinants and Apply Cramer’s Rule

12. 3.7 – Evaluate Determinants and Apply Cramer’s Rule
Example 5:
Use Cramer’s rule to solve the linear system.