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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
