ch 5 system of equations

8/10/2019 Ch 5 System of Equations

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MAY 2014 1WS Chapter 5 System of Equations

SOLVING SYSTEM OF

EQUATION


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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)


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System of Linear Equations

1. Two variables system of linear equations

2. Three variables system of linear equations


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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


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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


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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.


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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 :


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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.


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PROPERTIES OF MATRICES

Suppose A, B and C are m by n matrices.


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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


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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 :


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IDENTITY MATRIX

An n x n square matrix whose diagonalentries are 1s , while all others are 0s , is

called the identity matrix .


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PROPERTY

If A is a n by n square matrix, then

Example :


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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


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Example : Let

SOLUTION


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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


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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


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MAY 2014 18WS Chapter 5 System of Equations


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The right half is now


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SOLVING SYSTEM OF EQUATIONS BYUSING INVERSE MATRIX

Example Solve the system of equations

SOLUTIONThe above system is equivalent to the matrixequation


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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


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In the previous example (refer slide 19),we showed that

Thus,


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Lets practice.Solve each system of equations using theinverse matrix.

1.1. 2.

ch 5 system of equations

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