Calculus Week4

Mira Musrini (MMI)

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PRETEST1. Determinant of a matrix can be calculated for any

size of matrices . Give your opinion of this statement. Please explain the reasons of your answer!

2. Determine the right solution of given system of linear equation:

2x1 + +3x3 = 1

3x1 - x2 +4x3 = 7

6x1 + x2 - x3 = 0

Augmented matrix from systems of linear equations

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the augmented matrix for the given system of linear equations is

Consider the system of linear equations:

The next slide show how to resolve this system of linear equations using elementary rows operation.

Elementary rows operations

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Elementary rows operations

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Elementary rows operations

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Elementary rows operations

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Elementary rows operations

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Elementary rows operations

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Gauss Eliminations

In the preceding section, we solved the given system of linear equations by reducing the augmented matrix to

This matrix is in reduced row-echelon form.

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3100

2010

1001

Gauss EliminationsThe properties that possesed by matrices in reduced row-echelon form are : a.If a row does not consist of zero , then the first non zero number in the row is 1.b. If there are any rows that consist entirely zeros, then they are grouped together at the bottom of the matrixc. In any two successive rows that do not consist entirely zeros, the leading 1 in the lower rows occurs farther to the right than leading 1 in the higher row.d. Each column that contains a leading 1 has zeros everywhere else.

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Gauss Eliminations

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Gauss Eliminations

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Gauss Eliminations

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a method for inverting matrices

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a method for inverting matrices

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inverse of matrices

inverse of matricesDefinition : if A is square matrix , and if a matrix B is the same size can be found that AB=BA=I then A is said to be invertible and B is called inverse of A

IBA

IAB

AB

10

01

31

52

21

53

and

10

01

21

53

31

52

cesin,31

52 of inversean is

21

53

Non invertible matrices

If there is an zero row in the right side of final result, then the matrix is not invertible.The next example will describe above situation.

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Non invertible matrices

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Thus the matrix A does not have an invers matrix

POST TEST

1. By using elementary rows operation , determine wether matrix A is invertible or not

452

301

143

A

2. if A is invetible , check the answer of number 1 by multiplications.