MATRICES Using matrices to solve Systems of Equations

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: Cramer’s Rule (uses determinants) Matrix Equations (uses inverse matrices)

Cramer’s Rule - 2 x 2 Cramer’s Rule relies on determinants Consider the system below with variables x and y:

Cramer’s 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.

Cramer’s Rule - 3 x 3 Consider the 3 equation system below with variables x, y and z:

Cramer’s 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.

Cramer’s Rule Not all systems have a definite solution. If the determinant of the coefficient matrix is zero, a solution cannot be found using Cramer’s 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.

Cramer’s Rule Example:3x - 2y + z = 9 Solve the systemx + 2y - 2z = -5 x + y - 4z = -2

Cramer’s Rule 3x - 2y + z = 9 x + 2y - 2z = -5 x + y - 4z = -2 The solution is (1, -3, 0)

Matrix Equations Step 1: Write the system as a matrix equation. A three equation system is shown below.

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.

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!

Matrix Equations Example: Solve the system 3x - 2y = 9 x + 2y = -5

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

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:

Matrix Equations