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Lesson 1
Subject: Brackets 

Respected Sir, Aslam u alikum, Sir in handouts i can't understand the line, the decimal equivalent of binary number 10011001 is... Sir, the coding is 1001001, how we can determine that we should take powers in ve order? Q#2: in the question, (0.11.....11(0011))2 why we use brackets to write the result e.g (0011) , Respected Sir, kindly guide me in this regards.. Your's obidiently, 

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Instructor's Reply: 

Dear student, indefinitely repeated pattern of numbers has been enclosed in brackets with base as subscript. 

Subject: Query 

Respected Sir
Asalam o alakium
Plz explain example discussed in the lecture decimal equivalent of binary number 10011001 is 1 × 20 + 0* 2 1 + 0 × 22 + 1 × 23 + 1× 24 + 0 × 25+ 0 ×26 + 1×27 and the answer is (1.1953125)10...... Sir plz explain how u come to that binary number 10011001 in decimal is in fraction .......... that u start from 1 × 20 ............ What is the clue
Thanks 

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Instructor's Reply: 

Dear student, in daily life we make use of numbers 0,1,2,…,9 which is decimal system. We need to express this decimal system into other forms too. In computers different number systems are used
The number systems commonly used in computers are
so we should also know that how a number in one system can be expressed into another system. You want to get decimal equivalent of binary number 10011001. For this you express given number in form amNm + am1Nm1 +...+a1N1+a0+a1N1+...amNm 1 × 20 + 0* 2 1 + 0 × 22 + 1 × 23 + 1× 24 + 0 × 25+ 0 ×26 + 1×27 = 1+0+0+1/8+1/16+0+1/128 =1+1/8+1/16+1/128 =1+0.125+0.0625+0.0078125 =1.1953125 As now you have converted into decimal so enclose this obtained result into brackets and put ‘10’ as base =(1.1953125)10


Subject: Binary equal to decimal fraction 

Respected sir, 

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Instructor's Reply: 

Dear student, you are converting a decimal fraction into binary system. Integers you getting in each step are arranged as: The integral parts from top to bottom are arranged from left to right after the fractional point to represent the binary form. 

Student's Message: 
Msg No. 546844 
Subject: equation 

sir aslamo alikum im saria i want to ask about the equation that is A_{mN}^{m }+ A_{M1}N^{M1} +...+A_{1}N^{1}+A_{0}+A_{1}N^{1}+...A_{M}N^{M THAT IS CONFUSING and what is that a and n and m thanx.} 

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Instructor's Reply: 

Dear student 

Student's Message: 
Msg No. 546773 
Subject: Hexadecimal over Binary,decimal,Octal And Octal over Hexadecimal ,decimal and Binary 
Respected Sir, 
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Instructor's Reply: 
Dear student, The binary numeral system, or base2 number system, represents numeric values using two symbols: 0 and 1. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers and computerbased devices such as mobile phones. 
Lesson 2
dear sir i can't understand Local truncation error. and 2nd question. rational numbers which is written as p/q what is p and q as we write a/b or x/y thank yo. plz explain. 
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Instructor's Reply: 
Dear Student, Truncation error is the error made by cutting an infinite sum and approximating it by a finite sum. For instance, the trigonometric function “the cosine function” as written in series form on page 1 of lecture 2 in topic LOCAL TRUNCATION ERROR. This series is called Taylor series. if we approximate this cos function by the first three nonzero term of its Taylor series, and find its sum. Of course this answer will be different from the sum of infinite terms in this Taylor series, so an error has occurred. the resulting error is a truncation error.
A rational number is any number that can be expressed as the fraction a/b of two integers, with the neumerator 'a' and denominator 'b' such that 'b' is not equal to zero. The set of all rational numbers is usually denoted by a boldface Q. 
DEAR SIR: 
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Instructor's Reply: 
Dear student, yes you are right. 
Lesson 3
Kindly sir, explain truncation error with proper numerical method 
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Instructor's Reply: 
Dear student, this question has been already replied in this MDB. Truncate means chop out or cut off. Please read the reply I already gave and try to grasp this concept that if a function has been expressed in infinte series form. This sum can be found by certain formula developed particulary for those series. As may be you don't know how to find sum of certain infinite series, but just here concentrate on the main concept " truncation error how results." Let this infinite series sum has been found by a certain formula and you get its answer. Now let you cut off this series infinite terms and try to approximate it by its 1st 2 or 3 or 4 etc terms. Ofcourse this finite series sum will be different from the exact sum. this gives rise truncation error. 
Dear Sir, 
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Instructor's Reply: 
Dear student, for roots location in given interval we are making use of intermediate value property, and as we using bisection method, in this method root is taken to be mid point of those two values of ‘x’ for which graph of function crosses xaxis. Let for x0 and x1, we having intermediate value property i.e. f(x0)f(x1)<0, then the next root x2= (x0+x1)/2 . similarly this process continues till the required number of iterations. 
plz explain about Local truncation error with proper numerical example i cannot understand about that concept 
Post Your Comments 
Instructor's Reply: 
Dear Student, Truncation error is the error made by cutting an infinite sum and approximating it by a finite sum. For instance, the trigonometric function “the cosine function” as written in series form on page 1 of lecture 2 in topic LOCAL TRUNCATION ERROR. This series is called Taylor series. if we approximate this cos function by the first three nonzero term of its Taylor series, and find its sum. Of course this answer will be different from the sum of infinite terms in this Taylor series, so an error has occurred. the resulting error is a truncation error. 
Subject: Root Of Equation 
Respected Sir, 
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Instructor's Reply: 
Dear student, a real number x will be called a solution or a root if it satisfies the equation, The root of a equation is the value of x for which the equation y=f(x)=0 
Lesson 4
I am much confused about bisection method ............ How i came to know in which values root lies ................ Plz explain in simple way with example step by step ...........
Thanks 
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Instructor's Reply: 
Dear student, use intermediate value property to locate two values a and b such that f(a) and f(b) have opposite signs then we use the fact that the root lie between these points ‘a’ and ‘b’. Tell me did you understand it ? One thing important that when in the statement of question you have been given the interval, means two values i.e. initial and end values (a,b) then you do not use intermediate value property in this 1st step as you see in example on page 4 by bisection method. So initial guess has been known to lie in this interval, but for getting next guess you now make use of intermediate value property. This is what actually you are to do to know that where next value of root is lying. You read again Intermediate value property and then example on page 4. If you still feel any difficulty then ask me. 
Subject: Regula Falsi Method 
Respected Sir
Whenever intervals are given then we take as x0 and x1 and start from n =1 and whenever interval is not given them we take x1 and x2 and start fron n = 2 ............ Am i right ?????
Thanks 
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Instructor's Reply: 
Dear student, there is no difference that either you name two values as x0 and x1 OR x1 and x2, either interval is given or not. But if it is mentioned in the statement the values already let suppose x0,x1 then ofcourse you will name next value x2 and so on 
Lesson no 6
Subject: newton raphson method 
dear sir how we can find f'(x)???? and f"(x)???? 
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Instructor's Reply: 
Dear student, f'^{'}(x) means 1st derivative of f(x) and f"(x) means 2nd derivative of f(x). 
Subject: under root 
how can we use under root formula in MS Excel plz write method 
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Instructor's Reply: 
Dear student, use function SQRT in exel, for example you want to compute square root of 16. For this enter formula =SQRT(16) and press enter to get answer. 
f(x4)=ln1.3037cos1.3037=1.24816 plz calculate all step i do' not understand this calculation 
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Instructor's Reply: 
Dear student, compute values ln1.3037 and cos1.3037 separately and then find their difference 
Lecture n0 7
lecture 6 example no ! x3x21 .ess question me f(2) and f''(2) have the same signs so x0=2 kesy ata hay f(2)=5 and f"(2)=12 hy plz define 
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Instructor's Reply: 
Dear student, if f(a) and f''(a) are of same sign then 'a' can be chosen as initial approximation. In this example you have f(2)=5 and f''(2) =12 as both values '+5' and '+12' are of same sign so x0=2 is selected as initial approximation. 
What is difference in secant Method and Muller's method explain it with easy example 
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Instructor's Reply: 
Dear student, The secant method is modified form of NewtonRaphson method .
In Muller’s method, f (x) = 0 is approximated by a second degree polynomial; that is by a quadratic equation that fits through three points in the vicinity of a root. The roots of this quadratic equation are then approximated to the roots of the equation f (x)= 0.This method isiterative in nature and does not require the evaluation of derivatives as in Newton Raphson method. 
example f(x)= 4xcos2x(x2)sqare 
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Instructor's Reply: 
Dear student, observe that at end points '0' and '8' , values of f(0) and f(8) both have same negative sign. No change of sign so surely root can't lie in this interval and thats why as you do more iterations the value get more diverge. Observe x1=33.6801 and x2 = 38.8011 and x3=43.2343. Values not getting closer but diverging. So actually in this interval root has not been located. 
example f(x)= 4xcos2x(x2)sqare 
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Instructor's Reply: 
Dear student, observe that at end points '0' and '8' , values of f(0) and f(8) both have same negative sign. No change of sign so surely root can't lie in this interval and thats why as you do more iterations the value get more diverge. Observe x1=33.6801 and x2 = 38.8011 and x3=43.2343. Values not getting closer but diverging. So actually in this interval root has not been located. 
what is partial pivoting please explain it with easy way 
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Instructor's Reply: 
Dear student, You have to find solution of system of equations by Gaussian elimination method. Let suppose you have system of three equations.You are to make selection for diagonal elements in a special way , it is 1st you have diagonal element at position R1C1 means in 1st row and 1st column. For this check entries in C1. For this you are to look for the entry that has largest absolute value in C1 and make it as the pivot by interchanging two rows. (if there are more than one one such entries having the same largest absolute value , then select the one which is uppermost)). Next pivot position will be R2C2. For this you will now check entries in C2 that has largest absolute value and do the same strategy as I told you above. Similarly for position R3C3.This strategy is called partial pivoting. 
in this example x3=43.2343 while acutal its value becomes 34.3679 
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Instructor's Reply: 
Dear student, observe that at end points '0' and '8' , values of f(0) and f(8) both have same negative sign. No change of sign so surely root can't lie in this interval and thats why as you do more iterations the value get more diverge. Observe x1=33.6801 and x2 = 38.8011 and x3=43. 
example page No. 37 2coshx sinx=1 
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Instructor's Reply: 
Dear student, check on your calculator, there will be a button typed on it 'hyp'. To compute the said value press 'hyp' then button named 'cos' and then value 0.4 and then press button'=' . You will get this answer 
in example x33x+1 second step take x2=0 , xo=0.33333 , x1= 0.5 
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Instructor's Reply: 
Dear student, the root obtained from 1st iteration is x=0.33333, then for proceeding for 2nd iteration this computed root in 1st iteration will be treated as x0 now and c=fo=f(x0)=f(0.33333)= 0.037046 
Salam Respected Sir, 
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Instructor's Reply: 
Dear student, A system of linear equations (or linear system) is a collection of linear equationsas involving the same set of variables. For example, is a system of three equations in the three variables x, y, z. 
Sir i hv not studied math at inter level and bachelor level so i m confused about direct solution given in handouts and lectures. dnt know how to find out derivatives and function so pls guide me wat i do for better understanding. last time helping material u upload pls also xplane each step how to do that, bundle of thanx for ur coperation best regards 
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Instructor's Reply: 
Dear student, there is still nothing to worry about. You are to put a little more effort to learn these terminologies. As far it seems that basic concepts about functions and differentiation are making you confuse. For this I suggest you very direct and short speedy solution that please listen and learn Lecture # 6, 7, 9 of MTH 101 completely and then come to lectures 16,17 of MTH101. In lecture 16, 17 you just need to learn the formulae given as theorem statements. You are just to learn the formulae NOT their proof. It will make you able to grasp your problems yet you facing. After listening and learning you can discuss it with me in a better way through subject email id. I'd really like to know that you got it well so start from today as I told you and discuss it with me. 
Subject: Question 
A.A 
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Instructor's Reply: 
A finite difference is a mathematical expression of the form f(x + b) − f(x + a).
Only three forms are commonly considered: forward, backward, and central differences.
Central difference is = 1/2[f(x+h/2)+f(xh/2)] 
Subject: mc120204018 
what is the mean to use shift E, operator? 
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Instructor's Reply: 
Dear student, A SHIFT OPERATOR, denoted by E is the operator which shifts the value at the next point with step h i.e., Ef(x)=f(x+h). 
what method of finite difference operators would me more effective. what reason behind this? 
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Instructor's Reply: 
Dear student, 1st of all you have been told different types of operators and thier inter relationship . These operators will actually be helpful when in coming lectuires for equally spaced data, the interpolation and differentiation formulae will make use of backward, forward or central differences . For example if we desire the derivative at a point near the beginning of a tabulated data we use Newton's formula involving forward differences and so on . These all you are going to learn in coming lectures so after that you will be able to get it better. 
mc120204018 
what is important one type in finite difference operator? regards. 
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Instructor's Reply: 
Dear student, there are different types of operators backward difference operator, forward diference operator, shift operator, central difference operator etc. you are being introduced about these. They all are important at their own. You can be asked to use particular one for a problem or problem data itself decides that which operator should be used . So it is not the question that which one is most important. 
Exact solution of 2/3 is not exists.
TRUE
FALSE
The Jacobi’s method is
A method of solving a matrix equation on a matrix that has ____ zeros along its main diagonal.
No
At least one
A 3 x 3 identity matrix have three and __________eigen values.
Same
Different
Eigenvalues of a symmetric matrix are all _______ .
Real
Complex
Zero
Positive
The Jacobi iteration converges, if A is strictly diagonally dominant.
TRUE
FALSE
Below are all the finite difference methods EXCEPT _________.
Jacobi’s method
Newton’s backward difference method
Stirlling formula
Forward difference method
If n x n matrices A and B are similar, then they have the same eigenvalues (with the same multiplicities).
TRUE
FALSE
If A is a nxn triangular matrix (upper triangular, lower triangular) or diagonal matrix, the eigenvalues of A are the diagonal entries of A.
TRUE
FALSE
The characteristics polynomial of a 3x 3
Identity matrix is __________, if x is the Eigen values of the given 3 x 3 identity matrix. Where symbol ^ shows power.
(X1)^3
(x+1)^3
X^31
X^3+1
Two matrices with the same characteristic polynomial need not be similar.
TRUE
FALSE
Bisection method is a
Bracketing method
Open method
Regula Falsi means
Method of Correct position
Method of unknown position
Method of false position
Method of known position
Eigenvalues of a symmetric matrix are all _________.
Select correct option:
Real
Zero
Positive
Negative
An eigenvector V is said to be normalized if the coordinate of largest magnitude is equal to zero.
Select correct option:
TRUE
FALSE
Exact solution of 2/3 is not exists.
Select correct option:
TRUE
FALSE
The GaussSeidel method is applicable to strictly diagonally dominant or symmetric ________ definite matrices A.
Select correct option:
Positive
Negative
Differences methods find the ________ solution of the system.
Select correct option:
Numerical
Analytical
The Power method can be used only to find the eigenvalue of A that is largest in absolute value—we call this Eigenvalue the dominant eigenvalue of A.
Select correct option:
TRUE
FALSE
The Jacobi’s method is a method of solving a matrix equation on a matrix that has no zeros along its ________.
Select correct option:
Main diagonal
Last column
Last row
First row
If A is a nxn triangular matrix (upper triangular, lower triangular) or diagonal matrix , the eigenvalues of A are the diagonal entries of A.
Select correct option:
TRUE
FALSE
A 3 x 3 identity matrix have three and different Eigen values.
Select correct option:
TRUE
FALSE
Newton Raphson method falls in the category of
Bracketing method
Open Method
Iterative Method
Indirect Method
Newton Raphson method is also known as
Tangent Method
Root method
Open Method
Iterative Method
Secant Method uses values for approximation
1
3
2
4
Secant Method is than bisection method for finding root
Slow
Faster
In Newton Raphson method
Root is bracketed
Root is not bracketed
Regula falsi method and bisection method are both
Convergent
Divergent
In bisection method the two points between which the root lies are
Similar to each other
Different
Not defined
Opposite
In which methods we do not need initial approximation to start
Indirect Method
Open Method
Direct Method
Iterative Method
Root may be
Complex
Real
Complex or real
None
In Regula falsi method we choose points that have signs
2 points opposite signs
3 points opposite signs
2 points similar signs
None of the given
In a bounded function values lie between
1 and 1
1 and 2
0 and 1
0 and 2
Newton Raphson method is a method which when it leads to division of number close to zero
Diverges
Converges
Which of the following method is modified form of Newton Raphson Method?
Regula falsi method
Bisection method
Secant method
Jacobi’s Method
Which 1 of the following is generalization of Secant method?
Muller’s Method
Jacobi’s Method
Bisection Method
NR Method
Secant Method needs starting points
2
3
4
1
Near a simple root Muller’s Method converges than the secant method
Faster
Slower
If S is an identity matrix, then
All are true
If we retain r+1 terms in Newton’s forward difference formula, we obtain a polynomial of degree  agreeing with at
r+2
r+1
R
R1
P in Newton’s forward difference formula is defined as
Octal numbers has the base
10
2
8
16
Newton’s divided difference interpolation formula is used when the values of the independent variable are
Equally spaced
Not equally spaced
Constant
None of the above
Given the following data
0 
1 
2 
4 

1 
1 
2 
5 
Value of is
1.5
3
2
1
If is approximated by a polynomial of degree n then the error is given by
Let denotes the closed interval spanned by . Then vanishes times in the interval .
N1
N+2
N
N+1
Differential operator in terms of forward difference operator is given by
Finding the first derivative of at =0.4 from the following table:

0.1 
0.2 
0.3 
0.4 

1.10517 
1.22140 
1.34986 
1.49182 
Differential operator in terms of will be used.
Forward difference operator
Backward difference operator
Central difference operator
All of the given choices
For the given table of values
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 

0.425 
0.475 
0.400 
0.452 
0.525 
0.575 
, using twopoint equation will be calculated as.............
0.5
0.5
0.75
0.75
In Simpson’s 1/3 rule, is of the form
►
►
►
While integrating, , width of the interval, is found by the formula.
None of the given choices
To apply Simpson’s 1/3 rule, valid number of intervals are.....
7
8
5
3
For the given table of values
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 

0.425 
0.475 
0.400 
0.452 
0.525 
0.575 
, using threepoint equation will be calculated as ……
17.5
12.5
7.5
12.5
To apply Simpson’s 1/3 rule, the number of intervals in the following must be
2
3
5
7
To apply Simpson’s 3/8 rule, the number of intervals in the following must be
10
11
12
13
If the root of the given equation lies between a and b, then the first approximation to the root of the equation by bisection method is ……
None of the given choices
............lies in the category of iterative method.
Bisection Method
Regula Falsi Method
Secant Method
All of the given choices
For the equation, the root of the equation lies in the interval......
(1, 3)
(1, 2)
(0, 1)
(1, 2)
Rate of change of any quantity with respect to another can be modeled by
An ordinary differential equation
A partial differential equation
A polynomial equation
None of the given choices
If
Then the integral of this equation is a curve in
None of the given choices
Xtplane
Ytplane
Xyplane
In solving the differential equation
, By Euler’s method is calculated as
1.44
1.11
1.22
1.33
In second order RungeKutta method
is given by
None of the given choices
In fourth order RungeKutta method, is given by
In fourth order RungeKutta method, is given by
None of the given choices
AdamMoulton PC method is derived by employing
Newton’s backward difference interpolation formula
Newton’s forward difference interpolation formula
Newton’s divided difference interpolation formula
None of the given choices
The need of numerical integration arises for evaluating the definite integral of a function that has no explicit ____________ or whose antiderivative is not easy to obtain
Derivatives
Antiderivative
If then system will have a
Definite solution
Unique solution
Correct solution
No solution
If then
There is a unique solution
There exists a complete solution
There exists no solution
None of the above options
Direct method consists of method
2
3
5
4
We consider Jacobi’s method Gauss Seidel Method and relaxation method as
Direct method
Iterative method
Open method
All of the above
In Gauss Elimination method Solution of equation is obtained in
3 stages
2 stages
4 stages
5 stages
Gauss Elimination method fails if any one of the pivot values becomes
Greater
Small
Zero
None of the given
Changing the order of the equation is known as
Pivoting
Interpretation
Full pivoting is than partial pivoting
Easy
More complicated
The following is the variation of Gauss Elimination method
Jacobi’s method
Gauss Jordan Elimination method
Courts reduction method is also known as Cholesky Reduction method
True
False
Jacobi’s method is also known as method of Simultaneous displacement
True
False
Gauss Seidel method is also known as method of Successive displacement
False
True
In Jacobi’s method approximation calculated is used for
Nothing
Calculating the next approximation
Replaced by previous one
All above
In Gauss Seidel method approximation calculated is replaced by previous one
True
False
Relaxation method is derived by
South well
Not defined
Power method is applicable for only
Real metrics
Symmetric
Unsymmetrical
Both symmetric and real
The process of eliminating value of y for intermediate value of x is know as interpolation
True
False
20 MCQS
2 Questions of 2 marks
1) 1 question was to find the value of p from the given table using
Interpolation Newton's forward Diffrence Formula
2) we have to calculate the values of x, y and z for the given system of
equations using Gauss Siede iterative method taking initial solution vector
as (0, 0, 0)t
2 Questions of 3 marks
both of the questions were to find the value of theta for the given matrices
using Jacobi's method
2 Questions of 5 marks
1) Construct backward difference table for the given values
2) Find the solution of following system of Equations using Jacobi's
iterative method upto three decimal places
83x+11y4z=95
7x+52y+13z=104
3x+8y+29z=71
taking the initial solution vector as (0, 0, 0)T
Exact solution of 2/3 is not exists.
TRUE
FALSE
The Jacobi’s method is
a method of solving a matrix equation on a matrix that has ____ zeros along its main diagonal.
no
atleast one
A 3 x 3 identity matrix have three and __________eigen values.
same
different
Eigenvalues of a symmetric matrix are all _______ .
real
complex
zero
positive
The Jacobi iteration converges, if A is strictly diagonally dominant.
TRUE
FALSE
Below are all the finite difference methods EXCEPT _________.
jacobi’s method
newton's backward difference method
Stirlling formula
Forward difference method
If n x n matrices A and B are similar, then they have the same eigenvalues (with the same multiplicities).
TRUE
FALSE
If A is a nxn triangular matrix (upper triangular, lower triangular) or diagonal matrix , the eigenvalues of A are the diagonal entries of A.
TRUE
FALSE
The characteristics polynomial of a 3x 3
identity matrix is __________, if x is the eigen values of the given 3 x 3 identity matrix. where symbol ^ shows power.
(x1)^3
(x+1)^3
x^31
x^3+1
Two matrices with the same characteristic polynomial need not be similar.
TRUE
FALSE
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