2.2/2.3 - Solving Systems of Linear Equations
A Brief Introduction to MatricesMatrices are used to organize data efficiently and will help us to solve systems of linear equations. Matrices are usually denoted by a capital letter, say A. A matrix with m rows andn columns is called anm xnmatrix. This is the size (or dimension) of the matrix.
Ex: Size Name
A= " 2 3 # B = " 1 4 3 2 # C=h 1 5 7 i
To specify an element from a matrix, use its row and column number as subscripts of the lowercase letter of the matrix name.
(ROW NUMBER FIRST!)
Ex: Using the matrices above, find the following:
a11= a21= b12= b22=
5a21−3c13 =
Ex: Use a matrix to organize information from the following scenario:
Two brothers, Kevin and Mark, go to college. Kevin’s expenses include $800 for books, $4000 for tuition, and $2000 for a meal plan. Mark’s expenses include $1500 for a meal plan, $1200 for books, and $4500 for tuition.
Systems of Linear Equations and Matrices
A system of linear equations can be written as an augmented matrix. Ex: Write the following system of equations as an augmented matrix.
3x+ 6y−10z= 25
x+ 5z= 20−2y z= 15−2x
Ex: Write the given augmented matrix as an equivalent system of equations with variables x, y
and z. 1 −2 3 −1 3 4 −2 1 2 −3 7 6
You can useGauss-Jordan elimination (GJ) to solve any system of linear equations.
Idea: You will perform “operations” on your system that will result in an equivalent system where the answer to your system can easily be read. In order to simplify things, we will work with our system in matrix form and by using the “operations” we will be producing a matrix in “reduced row-echelon form” (the form where the answer can be easily found).
Gauss-Jordan Operations
1. Interchange any two equations (rows). Notation:
2. Replace an equation (row) by a NON-ZERO constant multiple of itself. Notation:
3. Replace an equation (row) by the sum of that equation (row) and a constant multiple of any other equation (row).
Notation:
Reduced Row-Echelon Form (RREF) of a Matrix
1. Each row consisting entirely of zeros lies below any other row having non-zero entries. 2. The first non-zero entry in each row is 1 (called aleading 1).
3. In any two succesive (non-zero) rows, the leading 1 in the lower row lies to the right of the leading 1 in the upper row.
4. If acolumn contains a leading 1, then the other entries in thatcolumn are zeros, called aunit column.
Ex: Which of the following matrices are in reduced row-echelon form? If the matrix is not in reduced form, which condition is violated?
(a) " 1 2 0 1 # (b) " 1 0 0 0 0 1 # (c) " 0 1 1 0 # (d) 1 0 0 0 0 0 0 0 1 (e) " −1 0 0 1 # (f) 1 2 1 0 0 0 0 0 0 (g) " 0 1 0 0 # (h) 1 0 3 0 1 0 0 0 0
Ex: Find the solution to the following system using Gauss-Jordan elimination: 3x+ 2y = 1
−7x−2y =−1 Solution:
STEP 1: Represent the system as anaugmented matrix.
STEP 2: Perform row operations on the augmented matrix until you have the left hand side of the matrix in reduced row-echelon form (Refer to Handout).
Ex: You are given the following system of equations:
x+ 2y−3z= 8 3y+ 4z= 10 2x−5z= 15 (a) Write the system as an augmented matrix.
(b) On this matrix perform the first two row operations you should use to get the matrix in reduced row-echelon form, showing the resulting matrices.
(c) Is the final matrix in (b) in reduced row-echelon form?
There is a built-in function on your calculator that will put a matrix in reduced row-echelon form. To use this function, you must have a matrix where the number of rows is less than or equal to the number of columns. You may use this function to perform the Gauss-Jordan method (or to help solve any system of equations), unless you are specifically told to perform specific steps (and show the resulting matrices) in the Gauss-Jordan process.
Calculator Instructions
Call the rref function: MATRX → MATH↓ B:rref(
Call the matrix to be used: MATRX→ NAMES↓ 1:[A] (or any matrix) Make your answer exact: MATH↓ 1:Frac
Ex: Solve the following systems of equations: (a) 3x−9y+ 6z=−12 x−3y+ 2z=−4 8−2x=−6y+ 4z (b) 2x−2y= 6 2−x= 2y
(c) x−2y+z=−3 −2z+ 2x+y= 2 x−3z+ 3y= 5 (d)x1+ 2x2+ 4x3 = 2 x1+x2+ 2x3 = 14 Note:
Number of Equations≥Number of Variables
=⇒ Any type of solution is possible (one, none, or infinitely-many solutions) Number of Equations<Number of Variables
Ex: A school has allocated $250,000 to purchase 100 new computers. The school has decided to purchase an assortment of desktops, small laptops, and high-end laptops costing $2000, $3000, and $5000 each, respectively. Find all options available to the school.
