PYQs

1) Explain the main steps of the Gauss-Jordan method and apply this method to find the inverse of the matrix \(\begin{bmatrix}{2} & {6} & {6} \\ {2} & {8} & {6} \\ {2} & {6} & {8}\end{bmatrix}\).

[2017, 10M]

2) Find the values of the two-valued variables \(A\), \(B\), \(C\) and \(D\) by solving the set of simultaneous equations \({\quad A^{\prime}+A . B=0} \\ {\quad A . B=A . C} \\ {\quad \text {A.B}+A . C^{\prime}+C . D=C^{\prime} . D}\).

[2001, 15M]

1) Apply Gauss-Seidel iteration method to solve the following system of equation:
\(2x+y-2z=17\)
\(3x+20y-z=-18\)
\(2x-3y+20z=25\), correct to three decimal places.

[2019, 15M]

2) Find the solution of the system
\(10 x_{1}-2 x_{2}-x_{3}-x_{4}=3\)
\(-2 x_{1}+10 x_{2}-x_{3}-x_{4}=15\) \(-x_{1}-x_{2}+10 x_{3}-2 x_{4}=27\)
\(-x_{1}-x_{2}-2 x_{3}+10 x_{4}=-9\)
using Gauss-Seidel method (make four iterations).

[2015, 15M]

3) Solve the system of equations \(2 x_{1}-x_{2}=7\)
\(-x_{1}+2 x_{2}-x_{3}=1\)
\(-x_{2}+2 x_{3}=1\)
using Gauss-Seidel iteration method (perform three iterations).

[2014, 15M]

4) Solve the following system of simultaneous equations, using Gauss-Seidel iterative method:

\(3 x+20 y-z=-18\)
\(20 x+y-2 z=17\)
\(2 x-3 y+20 z=25\)

[2012, 20M]

5) Given the system of equations:

\(2x+3y=1\)
\(2x+4y+z=2\)
\(2x+6z+Aw=4\)
\(4z+Bw=C\)

State the solvability and uniqueness conditions for the system. Give the solution when it exists.

[2010, 20M]

6) The equation \(x^{2}+a x+b=0\) has two real roots \(\alpha\) and \(\beta\). Show that the iterative method given by: \(x_{k+1}=-\dfrac{\left(a x_{k}+b\right)}{x_{k}}\), \(k=0,1,2 \ldots\) is convergent near \(x=\alpha\), if \(\vert \alpha \vert > \vert \beta \vert\).

[2009, 6M]

7) Apply Gauss-Seidel method to calculate \(x\), \(y\), \(z\) from the system:
\(-x-y+6 z=42\)
\(6 x-y-z=11.33\)
\(-x+6 y-z=32\)
with initial values (4.67, 7.62, 9.05). Carry out computations for two iterations.

[2008, 15M]

8) Using Gauss-Seidel iterative method, find the solution of the following system:
\(4 x-y+8 z=26\) \(5 x+2 y-z=6\) \(x-10 y+2 z=-13\)
up to three iterations.

[2004, 15M]

9) Using Gauss Seidel iterative method and the starting solution \(x_{1}=x_{2}=0\), determine the solution of the following system of equations in two iterations:
\(10 x_{1}-x_{2}-x_{3}=8\)
\(x_{1}+10 x_{2}+x_{3}=12\)
\(x_{1}-x_{2}+10 x_{3}=10\)

Compare the approximate solution with the exact solution.

[2001, 30M]