A square array of numbers or variables enclosed between vertical lines is called a
determinant. A determinant is different from a matrix in that a determinant has a numerical value, whereas a matrix does not. The following determinant has two rows and two columns.
The value of this determinant is found by finding the difference between the diagonally down product and the diagonally up product:
Example 1: Evaluate the following determinant.
Example 2: Solve the following system by using determinants.
To solve this system, three determinants will be created. One is called the
denominator determinant, labeled
D; another is the
x-numerator determinant, labeled
Dx; and the third is the
y-numerator determinant, labeled
Dy.
The denominator determinant,
D, is formed by taking the coefficients of
x and
y from the equations written in standard form.
The
x-numerator determinant is formed by taking the constant terms from the system and placing them in the
x-coefficient positions and retaining the
y-coefficients.
The
y-numerator determinant is formed by taking the constant terms from the system and placing them in the
y-coefficient positions and retaining the
x-coefficients.
The answers for
x and
y are
The check is left to you. The solution is
x = –5,
y = –2.
Many times, finding solutions by using determinants is referred to as
Cramer's rule, named after the mathematician who devised this method.
Example 3: Use Cramer's rule to solve this system.
The check is left to you. The solution is
x = ¼,
y = ⅓.
Linear Sentences in One Variable
Linear Sentences In Two Variables
