MATRICES

1
MATRICES

• Using matrices to solve Systems of Equations

2
Solving Systems with Matrices

• We can use matrices to solve systems that involve
  2 x 2 (2 equations, 2 variables) and 3 x 3 (3
  equations, 3 variables) systems. We will look at
  two methods

• Cramers Rule (uses determinants)
• Matrix Equations (uses inverse matrices)

3
Cramers Rule - 2 x 2

• Cramers Rule relies on determinants
• Consider the system below with
• variables x and y

4
Cramers Rule - 2 x 2

• The formulae for the values of x and y are shown
  below. The numbers inside the determinants are
  the coefficients and constants from the equations.

5
Cramers Rule - 3 x 3

• Consider the 3 equation system below with
  variables x, y and z

6
Cramers Rule - 3 x 3

• The formulae for the values of x, y and z are
  shown below. Notice that all three have the same
  denominator.

7
Cramers Rule

• Not all systems have a definite solution. If the
  determinant of the coefficient matrix is zero, a
  solution cannot be found using Cramers Rule
  because of division by zero.
• When the solution cannot be determined, one of
  two conditions exists
• The planes graphed by each equation are parallel
  and there are no solutions.
• The three planes share one line (like three pages
  of a book share the same spine) or represent the
  same plane, in which case there are infinite
  solutions.

8
Cramers Rule

• Example 3x - 2y z 9 Solve the system x
  2y - 2z -5 x y - 4z -2

9
Cramers Rule

• 3x - 2y z 9 x 2y - 2z
  -5 x y - 4z -2

The solution is (1, -3, 0)
10
Matrix Equations

• Step 1 Write the system as a matrix
  equation. A three equation
• system is shown below.

11
Matrix Equations

• Step 2 Find the inverse of the
  coefficient matrix.
• This can be done by hand for a 2 x 2
• matrix most graphing calculators can find the
  inverse of a larger matrix.

12
Matrix Equations

• Step 3 Multiply both sides of the matrix
• equation by the inverse.
• The inverse of the coefficient matrix times
• the coefficient matrix equals the identity
  matrix.

Note The multiplication order on the right side
is very important. We cannot multiply a 3 x 1
times a 3 x 3 matrix!
13
Matrix Equations

• Example Solve the system 3x - 2y 9 x
  2y -5

14
Matrix Equations

Multiply the matrices (a 2 x 2 times a 2 x
1) first, then distribute the scalar.
15
Matrix Equations

• Example 2 Solve the 3 x 3 system 3x - 2y z
  9 x 2y - 2z -5 x y - 4z -2

Using a graphing calculator
16
Matrix Equations