A determinant is a function whose input is a square matrix and whose output is a number. If A is a square matrix, then the determinant of A is represented by either | A | or
det ( A ). Do not confound the determinant symbol with absolute value. A determinant can be a negative number.

For a 2 X 2 matrix, the determinant is

.

Exercise 7.5.1

Find the determinant

Solution

For a 3 X 3 matrix, the determinant is

Exercise 7.5.2

Find the determinant

Solution

Cramer's Rule utilizes the coefficient matrix of a system of linear equations and determinants to find the solution.

Consider the general system of two linear equations in two variables:

a₁x + b₁y = c₁
a₂x + b₂y = c₂

The coefficient matrix is

.

Three determinants are defined in terms of the coefficient matrix:

Then Cramer's Rule finds the solution of the system of equations as

x = D_x / D
y = D_y / D

Exercise 7.5.3

Use Cramer's Rule to solve the system
 

  5 x + 3 y = - 2
 -3 x +   y =   4

Consider the general system of three linear equations is three variables:

a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃

The coefficient matrix is

_.

_{Four determinants are defined in terms of the coefficient matrix:}

Cramer's Rule finds the solution of the system of equations as

x = D_x / D
y = D_y / D
z = D_z / D

Exercise 7.5.4

Use Cramer's Rule to solve the system
 

2 x - 5 y + 2 z =  0
  x + 3 y +   z = -1
        y - 3 z =  0

Return