# Matrix Inversion Questions and Answers – Gauss Seidel Method

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This set of Numerical Methods Multiple Choice Questions & Answers (MCQs) focuses on “Gauss Seidel Method”.

1. Which of the following systems of linear equations has a strictly diagonally dominant coefficient matrix?
a)

3x – y = -4
2x + 5y = 2

b)

2x + y = 1
x - 7y = 4

c)

3x + 5y = 2
x + y = -3

d)

4x = 2y – z - 1
x + z = -4
3x – 5y + z = 3
Explanation: The coefficient matrix
A = $$\begin{bmatrix}3&-1\\2&5\end{bmatrix}$$
is strictly diagonally dominant because |3| > |-1| and |5| > |2|.

2. Gauss Seidal method is also termed as a method of _______
a) Successive displacement
b) Eliminations
c) False positions
d) Iterations

Explanation: Gauss Seidal method is also termed as a method of successive displacement because with the iterations, we keep on displacing the values we select for the unknowns which are used in the subsequent steps.

3. Gauss seidal method is similar to which of the following methods?
a) Iteration method
b) Newton Raphson method
c) Jacobi’s method
d) Regula-Falsi method

Explanation: The procedure involved in Gauss seidal and Jacobi is almost same. The only difference is that the while selecting the values for the unknowns, they use different methodology.

4. What is the main difference between Jacobi’s and Gauss-seidal?
a) Computations in Jacobi’s can be done in parallel but not in Gauss-seidal
b) Convergence in Jacobi’s method is faster
c) Gauss seidal cannot solve the system of linear equations in three variables whereas Jacobi cannot
d) Deviation from the correct answer is more in gauss seidal

Explanation: Computations in Jacobi’s can be done in parallel but not in Gauss-seidal because in Jacobi’s method, the entire set of values obtained during the previous iteration is used as it is in the next one, whereas in Gauss-seidal method, as we keep on getting the individual values of the variable, we use them in the subsequent steps.

5. While solving by Gauss Seidal method, which of the following is the first Iterative solution system; x – 2y = 1 and x + 4y = 4?
a) (1, 0.75)
b) (0.25,1)
c) (0,0)
d) (1,0.65)

Explanation: Here,
x – 2y = 1
x + 4y = 4
For first iteration we put n = 0 in the following equations,
xn+1 = 1 – 2yn
yn+1 = (1/4) (4 – xn+1)
We get,
x1 = 1 – 2(0) = 1
y1 = (1/4) (4 – 1) = 0.75.

6. The Gauss-Seidel method is applicable to strictly diagonally dominant or symmetric________ definite matrices.
a) Positive
b) Negative
c) Zero
d) Equal

Explanation: Gauss-Seidel method is applicable to strictly diagonally dominant or symmetric positive definite matrices because only in this case convergence is possible.

7. Gauss seidal requires less number of iterations than Jacobi’s method.
a) True
b) False

Explanation: Gauss-seidal requires less number of iterations than Jacobi’s method because it achieves greater accuracy faster than Jacobi’s method. This is the modification made to Jacobi’s method, which is now called as Gauss-seidal method.

8. Which of the following method is employed for solving the system of linear equations?
a) Runge Kutta
b) Newton Raphson
c) Gauss Seidal
d) Simpson’s Rule

Explanation: Runge Kutta method is used to solve differential equations. Newton-Raphson method is used to find the root of a polynomial. Simpson’s rule is used to solve integration problems. Gauss-seidal is used for solving system of linear equations.

9. What is the limitation of Gauss-seidal method?
a) It cannot be used for the matrices with non-zero diagonal elements
b) It is more complex than Jacobi’s method
c) It doesn’t guarantees convergence for each and every matrix
d) It is an iterative technique

Explanation: It does not guarantee convergence for each and every matrix. Convergence is only possible if the matrix is either diagonally dominant, positive definite or symmetric.

10. Solve the following equations by Gauss seidal method.

10a - 2b - c - d = 3
- 2a + 10b - c - d = 15
- a - b + 10c - 2d = 27
- a - b - 2c = 10d = -9

a) a = 1, b = 5, c = 7, d = -1
b) a = 1, b = 2, c = 3, d = 0
c) a = 2, b = 5, c = 3, d = 5
d) a = 2, b = 2, c = 8, d = -3

Explanation:
a = $$\frac{1}{10}$$(3 + 2b + c + d) …………….(i)
b = $$\frac{1}{10}$$(15 + 2a + c + d) ……………(ii)
c = $$\frac{1}{10}$$(27 + a + b +d) ……………..(iii)
d = $$\frac{1}{10}$$(-9 + a + b + 2c) ……………(iv)

First iteration
Putting b = 0, c = 0, d = 0 in (i), we get a = 0.3
Putting a = 0.3, c = 0, d = 0 in (ii), we obtain b = 1.56
Putting a = 0.3, b = 1.56, d = 0 in (iii), we obtain c = 2.886
Putting a = 0.3, b = 1.56, c = 2.886 in (iv), we get d = -0.1368

Second iteration
Putting b = 1.56, c = 2.886, d = -0.1368 in (i), we obtain a = 0.8869
Putting a = 0.8869, c = 2.886, d = -0.1368 in (ii), we obtain b = 1.9523
Putting a = 0.8869, b = 1.9523, d = -0.1368 in (iii), we have c = 2.9566
Putting a = 0.8869, b = 1.9523, c = 2.9566 in (iv) we get d = -0.0248

Third iteration
Putting b = 1.9523, c = 2.9566, d = -0.0248 in (i), we obtain a = 0.986
Putting a = 0.9836, c = 2.9566, d = -0.0248 in (ii), we obtain b = 1.9899
Putting a = 0.9836, b = 1.9899, d = -0.0248 i (iii), we get d = 2.9924
Putting a = 0.9836, b = 1.9899, c = 2.9924 in (iv), we get d = -0.0042

Fourth iteration proceeding as above
a = 0.9968 b = 1.9982, c = 2.9987, d = -0.0008
Fifth iteration is a = 0.9994, b = 1.9997, c = 2.9997, d = -0.0001
Sixth iteration is a = 0.9999, b = 1.9999, c = 2.9999, d = -0.0001
Hence, the solution is a = 1, b = 2, c = 3, d = 0.

11. Solve the following equation by Gauss Seidal Method up to 2 iterations and find the value of z.

27x + 6y - z = 85
6x + 15y + 2z = 72
x + y + 54z = 110

a) 1.88
b) 1.22
c) 0
d) 1.92

Explanation: From the given set of equations-
x=$$\frac{(85-6y+z)}{27}$$
y=$$\frac{(72-6x-2z)}{15}$$
y=$$\frac{(110-x-y)}{54}$$

1st iteration:
y=0, z=0
x=$$\frac{85}{27}$$=3.14
x=3.14, z=0
y=$$\frac{72-(6×3.14)-2×(0)}{15}$$=3.54
x=3.14, y=3.54
z=$$\frac{110-3.14-3.54}{54}$$=1.91

2nd iteration:
z=1.91, y=3.54
x=$$\frac{85-(6×3.54)+(1.91)}{27}$$=2.43
z=1.91, x=2.43
y=$$\frac{72-(6×2.43)-2×(1.91)}{15}$$=3.57
y=3.57, x=2.43
z=$$\frac{110-2.43-3.57}{54}$$=1.92
Thus, after the second iteration
x=2.43, y=3.57, z=1.92.

12. Solve the following equation by Gauss Seidal Method up to 3 iterations and find the value of x.

4x - 3y - z = 40
x - 6y + 2z = -28
x - 2y + 12z = -86

a) x=11.11
b) x=13.28
c) x=11.51
d) x=9.86

Explanation: From the given set of equations-
x=$$\frac{(40+3y+z)}{4}$$
y=$$\frac{(28+x+2z)}{6}$$
z=$$\frac{(-86-x+2y)}{12}$$
1st iteration:
y=0, z=0
x=$$\frac{40}{4}$$=10
x=10, z=0
y=$$\frac{28+(10)+2×(0)}{6}$$=6.33
x=10, y=6.33
z=$$\frac{-86-10+(2×6.33)}{12}$$=-6.94
2nd iteration:
z=-6.94, y=6.33
x=$$\frac{40+(3×6.33)+(-6.94)}{4}$$=13.01
z=-6.94, x=13.01
y=$$\frac{28+(13.01)+2×(-6.94)}{6}$$=4.52
y=4.52, x=13.01
z=$$\frac{-86-13.01+(2×4.52)}{12}$$=-7.49
3rd iteration:
z=-7.49, y=4.52
x=$$\frac{40+(3×4.52)+(-7.49)}{4}$$=11.51
z=-7.49, x=11.51
y=$$\frac{28+(11.51)+2×(-7.49)}{6}$$=4.08
y=4.08, x=11.51
z=$$\frac{-86-11.51+(2×4.08)}{12}$$=-7.44
Thus, after the third iteration
x=11.51, y=4.08, z=-7.44.

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