Solving a system of linear equations Let A be a matrix, X a column vector, B a column vector then the system of linear equations is denoted by AX=B.

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1 Matries and Vetors: Leture Solving a sstem of linear equations Let be a matri, X a olumn vetor, B a olumn vetor then the sstem of linear equations is denoted b XB. The augmented matri The solution to a sstem of linear equations suh as epends on the oeffiients of and and the onstants on the right-hand side of the equation. The matri of oeffiients for this sstem is the matri If we insert the onstants from the right-hand side of the sstem into the matri of oeffiients, we get the matri We use a vertial line between the oeffiients and the onstants to represent the equal signs. This matri is the augmented matri of the sstem also it an be written as: Note: Two sstems of linear equations are equivalent if the have the same solution set. Two augmented matries are equivalent if the sstems the represent are equivalent. E.: Write the augmented matri for eah sstem of equations. a) b)

2 Matries and Vetors: Leture We'll take two methods to solve the sstem XB ) Cramer's rule The solution to the sstem a b a b Is given b a a b, b and where b and b a a Provided that Notes:. Cramer's rule works on sstems that have eatl one solution.. Cramer's rule gives us a preise formula for finding the solution to an independent sstem.. Note that is the determinant made up of the original oeffiients of and. is used in the denominator for both and. is obtained b replaing the first or ) olumn of b the onstants and. is found b replaing the seond or ) olumn of b the onstants and. E.: Use Cramer's rule to solve the sstem: First find the determinants,, and : ) - -, B Cramer's rule, we have and Chek in the original equations. The solution set is, ).

3 Matries and Vetors: Leture E.: Solve the sstem: 9 Cramer's rule does not work beause ) Beause Cramer's rule fails to solve the sstem, we appl the addition method: 9 Beause this last statement is false, the solution set is empt. The original equations are inonsistent. E.: Solve the sstem: Cramer's rule does not appl beause ) Multipl Eq.) b - and add it to Eq.) Beause the last statement is an identit, the equations are dependent. The, ). solution set is { } E.: Use Cramer's rule to solve the sstem: ) First write the equations in standard form, B C Find,, and : 9 - -, Using Cramer's rule, we get and Beause,) satisfies both of the original equations, the solution se is {,)}.

4 Matries and Vetors: Leture ) The Gaussian Elimination method When we solve a single equation, we write simpler and simpler equivalent equations to get an equation whose solution is obvious. In the Gaussian elimination method we write simpler and simpler equivalent augmented matries until we get an augmented matri in whih the solution to the orresponding sstem is obvious. Beause eah row of an augmented matri represents an equation, we an perform the row operations on the augmented matri. Elementar ow Operation:. Construt the augmented matri :B).. Interhange two rows i j ).. Multipl an row b a onstant different from ero i k i ). dd a onstant multipl of an row to another row i i k j ) E.: Use Gaussian elimination method to solve the sstem two equations in two variables): Start with the augmented matri: - This augmented matri represents the sstem and. So the solution,. set to the sstem is { )} E.: Use Gaussian elimination method to solve the sstem three equations in three variables):

5 Matries and Vetors: Leture This augmented matri represents the sstem, and. So the solution set to the sstem is ) { },,. E.: Solve the sstem orresponds to the equation. So the equations are inonsistent, and there is no solution to the sstem. Matri Inverse The matri has an inverse denoted b - if where. - I. We'll take two methods to find - where is an n n matri. ) B Gauss elimination methodusing row operations):. Construt the augment matri :I). Use row operation until we have I: - ) E: Use ow operation to find - if

6 Matries and Vetors: Leture - E: Find - if - - -

7 Matries and Vetors: Leture ) B Cofator Method Using determinant of the matri) The ofator of the element a ij of the matri a ij ) is defined b ij -) ij ij where ij is the determinant of the matri that remains when the row i and the olumn j are deleted. To find the inverse of a matri whose determinant is not ero - onstrut the matri of ofators of, of ) ij - Construt the transposed matri of ofators alled the adjoin of adj ) of ) T - then - ) det adj - to hek our answer. - I or -. I E.: Use determinant to find - where - ) adj Cof) C -) C -) - C -) - C -) dj) T E: Find - if Solution:

8 Matries and Vetors: Leture ) of, ) ),,,, ),, ), t of dj)