8/10/2019 Ch 5 System of Equations
1/23
MAY 2014 1WS Chapter 5 System of Equations
SOLVING SYSTEM OF
EQUATION
8/10/2019 Ch 5 System of Equations
2/23
MAY 2014 WS Chapter 5 System of Equations 2
LEARNING OBJECTIVES :
Be able to solve the system of linear equations
using:Inverse matrixCramers Rule
Gaussian Elimination Method(Reduced Row Echelon Form)
8/10/2019 Ch 5 System of Equations
3/23
MAY 2014 WS Chapter 5 System of Equations 3
System of Linear Equations
1. Two variables system of linear equations
2. Three variables system of linear equations
8/10/2019 Ch 5 System of Equations
4/23
MAY 2014 WS Chapter 5 System of Equations 4
Types of solutions .
2. Infinitely many solutionsGraphically, when the lines representing
the equations overlap each otherBy solving, we obtained an identity 0 = 0
1. Unique solution
3. No solution
Graphically, when the lines representingthe equations are parallel to each other.By solving, we obtained a false statement ,0 = 4
8/10/2019 Ch 5 System of Equations
5/23
MAY 2014 WS Chapter 5 System of Equations 5
MATRIX
A matrix is a rectangular array of numberswritten within brackets.
Column
R o w3 by 3 matrix
Each number in the matrix is called anelement of the matrix.is the row index and is the column index
8/10/2019 Ch 5 System of Equations
6/23
MAY 2014 WS Chapter 5 System of Equations 6
SQUARE MATRIXA matrix A with n rows and n columns is calleda square matrix.i.e : number of columns = number of rowsEg : 2 x 2, 3 x 3, 4 x 4
ZERO MATRIXThe matrix with all entries 0.
8/10/2019 Ch 5 System of Equations
7/23
MAY 2014 WS Chapter 5 System of Equations 7
ADDITION & SUBTRACTION OF MATRICES
If A and B are two matrices of the same size ,then the sum A+B is obtained by adding thecorresponding entries in the two matrices
Note :A + B = B + A
Subtraction of matrix is the same as Addition
Note :
8/10/2019 Ch 5 System of Equations
8/23
MAY 2014 WS Chapter 5 System of Equations 8
SCALAR MATRIXIf any matrix is multiplied with any scalar c,then the product cA is the matrix obtained
by multiplying each entry of A by c.
8/10/2019 Ch 5 System of Equations
9/23
MAY 2014 WS Chapter 5 System of Equations 9
PROPERTIES OF MATRICES
Suppose A, B and C are m by n matrices.
8/10/2019 Ch 5 System of Equations
10/23
MAY 2014 WS Chapter 5 System of Equations 10
MULTIPLICATION
If A is a m x r matrix and B is r x n matrix,then the product AB is the m x n matrix.
A
m by r
B
r by n
Must be the same
for AB to be definedAB is m by n
8/10/2019 Ch 5 System of Equations
11/23
MAY 2014 WS Chapter 5 System of Equations 11
NOTE : Before multiplying any matrices,always check
Column of A = row of BIf this is not satisfied, then A x B is undefined.
Example : MULTIPLYING TWO MATRICES
Col A = Row B
2 x 3 3 x 22 x 2
Note :
8/10/2019 Ch 5 System of Equations
12/23
MAY 2014 WS Chapter 5 System of Equations 12
IDENTITY MATRIX
An n x n square matrix whose diagonalentries are 1s , while all others are 0s , is
called the identity matrix .
8/10/2019 Ch 5 System of Equations
13/23
MAY 2014 WS Chapter 5 System of Equations 13
PROPERTY
If A is a n by n square matrix, then
Example :
8/10/2019 Ch 5 System of Equations
14/23
MAY 2014 WS Chapter 5 System of Equations 14
INVERSE OF A MATRIX
If there exists an n x n matrix with theproperty that
Then we say that is the inverse of A .
Let A be a square n x n matrix.
INVERSE OF 2 x 2 MATRIX
8/10/2019 Ch 5 System of Equations
15/23
MAY 2014 WS Chapter 5 System of Equations 15
Example : Let
SOLUTION
8/10/2019 Ch 5 System of Equations
16/23
MAY 2014 WS Chapter 5 System of Equations 16
FINDING THE INVERSE OF MATRIX A
The Procedure :Step 1 : Form the augmented matrix
Step 2 : Transform into reducedechelon form
Step 3 : The reduced echelon form ofwill contain on the left bar of the
vertical bar ; the n by n matrix onthe right is the
8/10/2019 Ch 5 System of Equations
17/23
MAY 2014 WS Chapter 5 System of Equations 17
Example : The matrix
is non singular. Find its inverse.SOLUTION
Step 1 : Form the augmented matrix
Then we use the row operations to transformto the echelon form
8/10/2019 Ch 5 System of Equations
18/23
MAY 2014 18WS Chapter 5 System of Equations
8/10/2019 Ch 5 System of Equations
19/23
MAY 2014 WS Chapter 5 System of Equations 19
The right half is now
8/10/2019 Ch 5 System of Equations
20/23
MAY 2014 WS Chapter 5 System of Equations 20
SOLVING SYSTEM OF EQUATIONS BYUSING INVERSE MATRIX
Example Solve the system of equations
SOLUTIONThe above system is equivalent to the matrixequation
8/10/2019 Ch 5 System of Equations
21/23
MAY 2014 WS Chapter 5 System of Equations 21
We solve this matrix equation by multiplyingeach side by the inverse of A
Multiply both sideswith
Associative property
Property of inverses
Property of identity matrix
8/10/2019 Ch 5 System of Equations
22/23
MAY 2014 WS Chapter 5 System of Equations 22
In the previous example (refer slide 19),we showed that
Thus,
8/10/2019 Ch 5 System of Equations
23/23
MAY 2014 WS Chapter 5 System of Equations 23
Lets practice.Solve each system of equations using theinverse matrix.
1.1. 2.