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BCA Semester 4 – Numerical Methods – Unit 4: Solution of Linear Algebraic Equations (10Hrs.)

Comprehensive questions and detailed answers for Unit 4: Solution of Linear Algebraic Equations (10Hrs.). Perfect for exam preparation and concept clarity.

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1

What is meant by ill-conditioned system? Find the Cholesky decomposition of the following matrix:

\quad \quad \quad \begin{bmatrix} 4 & 1 & 1 \\ 1 & 5 & 2 \\ 1 & 2 & 3 \end{bmatrix}

MediumTHEORY5 marks2021(TU FOHSS Final)

2

a) Solve the following system of equations using Gauss-Seidel method:

\quad\quad\quad 2x - 7y - 10z = -175

\quad\quad\quad x + y + 3z = 14

\quad\quad\quad x + 10y + 9z = 7

b) Write an expansion of Taylor’s Theorem. Solve the equation

\quad\quad\quad y′(x)=x2+y2, \quad y(0)=1

for $x=0.5$ using Taylor method.

MediumTHEORY10 marks2021(TU FOHSS Final)

3

Find the dominant eigenvalue of

A = \begin{bmatrix} 2 & 3 \\ 5 & 4 \end{bmatrix}

by the power method. Apply the Gauss--Seidel method to solve the system of equations:

\begin{aligned} 6x_1 - 2x_2 + x_3 &= 11, \\ -2x_1 + 7x_2 + 2x_3 &= 5, \\ x_1 + 2x_2 - 5x_3 &= -1, \end{aligned}

with the initial vector of $(0, 0, 0).$

MediumTHEORY10 marks2022(TU FOHSS Final)

4

What is the form of resultant matrix using Gauss-Jordan method? Solve the following system of equations using Gauss-Jordan Method.

\quad \quad  x + 2y - 3z = 4

\quad \quad2x + 4y - 6z = 8

\quad \quad  x - 2y - 5z = 4

MediumTHEORY5 marks2023(TU FOHSS Final)

5

Solve the following equations by using Gauss-Jordan method.

\quad \quad2x+3y+4z=5

\quad \quad3x+4y+5z=6

\quad \quad4x+5y+6z=7

MediumTHEORY5 marks2024(TU FOHSS Final)

6

Solve the given set of linear equations using Dolittle LU decomposition method:

\quad \quad3x 1 + 2x 2 + x 3 = 10

\quad \quad2x 1 + 3x 2 + 2x 3 = 14

\quad \quad3x 1 + 2x 2 + 3x 3 = 14

MediumTHEORY10 marks2024(TU FOHSS Final)

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Sample Questions

What is meant by ill-conditioned system? Find the Cholesky decomposition of the following matrix: \[ \quad \quad \quad \begin{bmatrix} 4 & 1 & 1 \\ 1 & 5 & 2 \\ 1 & 2 & 3 \end{bmatrix} \]

Marks: 5Chapter: Unit 4: Solution of Linear Algebraic Equations (10Hrs.)

a) Solve the following system of equations using Gauss-Seidel method: \[ \quad\quad\quad 2x - 7y - 10z = -175 \] \[ \quad\quad\quad x + y + 3z = 14 \] \[ \quad\quad\quad x + 10y + 9z = 7 \] b) Write a

Marks: 10Chapter: Unit 4: Solution of Linear Algebraic Equations (10Hrs.)

Find the dominant eigenvalue of \[ A = \begin{bmatrix} 2 & 3 \\ 5 & 4 \end{bmatrix} \] by the power method. Apply the Gauss--Seidel method to solve the system of equations: \[ \begin{aligned} 6x1 - 2x

Marks: 10Chapter: Unit 4: Solution of Linear Algebraic Equations (10Hrs.)

What is the form of resultant matrix using Gauss-Jordan method? Solve the following system of equations using Gauss-Jordan Method. \[ \quad \quad x + 2y - 3z = 4 \] \[ \quad \quad2x + 4y -

Marks: 5Chapter: Unit 4: Solution of Linear Algebraic Equations (10Hrs.)

Solve the following equations by using Gauss-Jordan method. \[ \quad \quad2x+3y+4z=5 \] \[ \quad \quad3x+4y+5z=6 \] \[ \quad \quad4x+5y+6z=7 \]

Marks: 5Chapter: Unit 4: Solution of Linear Algebraic Equations (10Hrs.)

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