Title: 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

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