1. H.Melikian/12101 System of Linear Equations and Augmented Matrices Dr.Hayk Melikyan Departmen of Mathematics and CS melikyan@nccu.edu

2. H.Melikian/12102 It is impractical to solve more complicated linear systems by hand. Computers and calculators now have built in routines to solve larger and more complex systems. Matrices, in conjunction with graphing utilities and or computers are used for solving more complex systems. In this section, we will develop certain matrix methods for solving two by two systems.

3. H.Melikian/12103 Matrices v A matrix is a rectangular array of numbers written within brackets. Here is an example of a matrix which has three rows and three columns: The subscripts give the “address” of each entry of the matrix. For example the entry v Is found in the second row and third column

4. H.Melikian/12104 MATRICES A MATRIX is a rectangular array of numbers written within brackets. Each number in a matrix is called an ELEMENT. If a matrix has m rows and n columns, it is called an m x n MATRIX; m x n is the SIZE; m and n are the DIMENSION S. A matrix with n rows and n columns is a S QUARE MATRIX OF ORDER n. A matrix with only one column is a column matrix; a matrix with only one row is a ROW MATRIX. The element in the i th row and j th column of a matrix A is denoted a ij. The PRINCIPAL DIAGONAL of a matrix A consists of the elements a 11, a 22, a 33, ….

5. H.Melikian/12105 Matrix solutions of linear systems v When solving systems of linear equations, the coefficients of the variables played an important role. We can represent a linear system of equations using what is called an augmented matrix, a matrix which stores the coefficients and constants of the linear system and then manipulate the augmented matrix to obtain the solution of the system. Here is an example: v x +3y=5 v 2x – y=3

6. H.Melikian/12106 Generalization v Linear system: v Associated augmented matrix:

7. H.Melikian/12107 Operations that Produce Row-Equivalent Matrices: v 1. Two rows are interchanged: v 2. A row is multiplied by a nonzero constant: v 3. A constant multiple of one row is added to another row:

8. H.Melikian/12108 Solve using Augmented matrix: v Solve v x +3y=5 v 2x – y=3 v 1. Augmented system v 2. Eliminate 2 in 2 nd row by row operation v 3. Divide row two by -7 to obtain a coefficient of 1. v 4. Eliminate the 3 in first row, second position. v 5. Read solution from matrix v:v:

9. H.Melikian/12109 Solving a system using augmented matrix methods v x+2y=4 v x+(1/2)y=4 1. Eliminate fraction in second equation. 2. Write system as augmented matrix. 3. Multiply row 1 by -2 and add to row 2 4. Divide row 2 by -3 5. Multiply row 2 by -2 and add to row 1. 6. Read solution : x = 4, y = 0 7. (4,0)

10. H.Melikian/121010 Solving a system using augmented matrix methods 10x -2y=6 -5x+y= -3 v 1. Represent as augmented matrix. v 2. Divide row 1 by 2 v 3. Add row 1 to row 2 and replace row 2 by sum v 4. Since 0 = 0 is always true, we have a dependent system. The two equations are identical and there are an infinite number of solutions.

11. H.Melikian/121011 Another example v Solve v Rewrite second equation : v Since we have an impossible equation, there is no solution. The two lines are parallel and do not intersect.

12. H.Melikian/121012 Recall A system of linear equations is transformed into an equivalent system if: (a)two equations are interchanged; (b)an equation is multiplied by a nonzero constant; (c)a constant multiple of one equation is added to another equation

13. H.Melikian/121013 Given the system of linear equations (I) and its associated augmented matrix (II). If (II) is row equivalent to a matrix of the form