Question #2: Solve the system of linear equations with the help of Gaussian elimination method.

2x + y + z = 9;3x −2y + 4z = 9;x +y-2z = 3

System of Linear Equations entered :

[1] 2x + y + z = 9

[2] 3x - 2y + 4z = 9

[3] x + y - 2z = 3

Solve by Substitution :

// Solve equation [3] for the variable y

[3] y = -x + 2z + 3

// Plug this in for variable y in equation [1]

[1] 2x + (-x +2z+3) + z = 9

[1] x + 3z = 6

// Plug this in for variable y in equation [2]

[2] 3x - 2•(-x +2z+3) + 4z = 9

[2] 5x = 15

// Solve equation [2] for the variable x

[2] 5x = 15

[2] x = 3

// Plug this in for variable x in equation [1]

[1] (3) + 3z = 6

[1] 3z = 3

// Solve equation [1] for the variable z

[1] 3z = 3

[1] z = 1

// By now we know this much :

x = 3

y = -x+2z+3

z = 1

// Use the x and z values to solve for y

y = -(3)+2(1)+3 = 2

Solution :

{x,y,z} = {3,2,1}

Convert the decimal number 80 into its binary equivalent.

Question 2:

Convert the binary number 2 (11001100) to its decimal equivalent.

Question 3:

Find the relative error when 17 is considered upto four decimal places.

Question 4:

Find the interval in which atleast one root of the equation 3 2 xx x 2 10 lies.

Question 5:

Find the real root of the equation 4 x x 10 0 in the interval [1, 2] by bisection method upto

two iterations. ]]>

Spring 2020_MTH603_1_SOL.docx ]]>

u(2)=11.4817⎛⎝⎜⎜0.3981760.8212671.0⎞⎠⎟⎟

we have the largest eigen value and the corresponding eigenvector as

]]>O(h2)

O(h3)

**O(h4)**

None of the given choices

If n x n matrices A and B are similar, then they have the same eigenvalues (with the same multiplicities).

True

False

Since similar matrices A and B have the same characteristic polynomial, they also have the same eigenvalues. If B = PAP−1 and v = 0 is an eigenvector of A (say Av = λv) then B(Pv) = PAP−1(Pv) = PA(P−1P)v = PAv = λPv. Thus Pv (which is non-zero since P is invertible) is an eigenvector for B with eigenvalue λ.

]]>Question #1: Find the root of the equation, Perform three iteration of the equation,

ln (x −1) + sinx =0 by using Newton Raphson method.

Ans: Let f(x) = ln(x+1) + sinx = 0 and f(x) = 1/(x-1) + cosx

F (1.5) = ln(0.5) + (1.5) = - 0.0667

F(2) = ln(1) + sin(2) = 0.035

Since f (1.5) f (2) < 0 so roots lies in interval [1.5, 2]

Let x0 = 1.75 . x0 can be taken in the interval any real number [ 1.5 , 2 ], we let mid point

of this interval .

As we know Newton Raphson method is

Xn+1 = xn – f ( xn ) / f(xn)

First iteration

X1 = x0 –f(x0) / f(x0) = 1.75 - f(1.75) / f(1.75)

= 1.75 – (-0.2571 / 2.3329) = 1.8602

Second iteration:

X2 = x1 - f(x) / f(x) = 1.8602 –[ f(1.8602) / f(1.8602)]

= 1.8602 - ( -0.1181 / 2.1620 ) = 1.9148

Third iteration:

X3 = x2- f(x2) / f(x2) = 1.9148 –f(1.9148) / f(1.9148)

= 1.9148 – [-0.0556/2.0926]

= 1.9414

Question #2: Solve the system of linear equations with the help of Gaussian elimination method.

x + y + z = 6;2x − y + z = 3;x + z = 4

ANS: In Gaussian elimination method we convert the augmented matrix into reduce

Echelon form therefore,

Augmented matrix is

R2- 2R1 , R3 – R1

-1R2 , -1R3

R23

R3-3R2

X + Y+ Z = 6 ;………………….(1)

Y = 2,

Z = 3

Put into eq (1),

we get X = 1 ,