Math Calculators, Lessons and Formulas

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Determinants: (lesson 2 of 2)

## Applications of Determinants

### Cramer's rule:

For linear system , if , then the system has the unique solution,

where is the matrix obtained by replacing the i-th column of A by b.

Example 1:

Solve for the following system of linear equations by Cramer's rule,

Solution:

The coefficient matrix A and the vector b are

,

respectively. Then,

Thus, .

### Areas:

Triangle:

Consider the triangle with vertices and .

The area of the triangle is

.

Example 2:

Compute the area of the triangle with vertices and .

Solution:

The area is

.

Parallelogram:

Suppose we have a parallelogram with vertices , and ,

Then, the area of the parallelogram is