Matrix Inversion Questions and Answers – Factorization Method

«
»

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

1. The LU method of factorization was introduced by the mathematician____________
a) Alan Tango
b) David Hilbert
c) G. W. Leibniz
d) Alex Grothendieck

Explanation: The LU method of factorization was introduced by the mathematician Alan Tango. He was an English mathematician.

2. Which of the following step is not involved in the factorization process?
a) converting the given system to matrix form
b) the matrix is decomposed into the product of lower and upper triangular matrix
c) finding the unknowns using matrix multiplication
d) elimination of unknowns using back substitution

Explanation: The first step is to convert the given system to matrix form. And then after using this method the matrix is decomposed into the product of lower and upper triangular matrix. Lastly, the unknowns are found using matrix multiplication.

3. Errors may occur in performing numerical computation on the computer due to which of the following reasons?
a) Rounding errors
b) Power fluctuation
c) Operator fatigue
d) Back substitution

Explanation: Rounding errors generally produces errors while performing numerical computation on the computer. Power fluctuation and operator fatigue are not the general problems which occur in performing numerical computations.

4. What is the other name for factorization method?
a) Doolittle’s Method
b) Lin Bairstow Method
c) Muller’s Method
d) Decomposition Method

Explanation: Another name for factorization method is Doolittle’s Method as Doolittle’s method is basically an algorithm of Factorization method.

5. Factorization can be viewed as matrix form of Gauss Elimination method.
a) True
b) False

Explanation: Factorization can be viewed as matrix form of Gauss Elimination method as it also involves the same procedure only the matrix solving part is different.

6. What is the principle of factorization?
a) There exists no inverse for a singular matrix
b) Determinant of an identity matrix is one
c) Every square matrix can be expressed as a product of a lower triangular matrix and upper triangular matrix
d) Every matrix can be expressed as a sum of a skew symmetric and a symmetric matrix

Explanation: The principle of factorization is that every square matrix can be expressed as a product of a lower triangular matrix and upper triangular matrix. On the basis of this fact, these lower and upper triangular matrices help us in finding the unknowns.

7. What is the condition applied in factorization method?
a) Matrix should not be singular
b) Back substitution should be done
c) There must exist a diagonal matrix form of the given matrix
d) All principal minors of the matrix should be non-singular

Explanation: The necessary condition for factorization method is that all principal minors of the matrix should be non-singular. Otherwise, there will be no formation of lower and upper triangular matrix.

8. During computation of elements of L and U, what order should be followed?
a) First row of U -> Second row of L -> Second column of U -> Second row of U -> Third row of U
b) First row of U -> First column of L -> Second row of U -> Second column of L -> Third row of U
c) First column of U -> First row of L -> Second column of U -> Second row of L -> Third row of U
d) First row of L -> First column of U -> Second row of L -> Second column of U -> Third column of L

Explanation: While solving for L and U, First row of U -> Second row of L -> Second column of U -> Second row of U -> Third row of U, this order should be followed. This is sequential arrangement of steps which easily helps in finding the unknowns.

9. What is the other application of Factorization method?
a) Finding inverse of a matrix
b) Solving partial differential equations
c) Solving ordinary differential equations
d) Finding Eigen values

Explanation: We can even find the inverse of a matrix using factorization method. This can be done by finding inverse of the intermediate matrices and then using them to find the inverse of the desirable matrix.

10. Factorization method is superior than which of the following methods?
a) Gauss Elimination
b) Gauss Seidal
c) Jacobi’s Method
d) Iterative Method

Explanation: Factorization method is superior to Gauss Elimination due to the involvement of less number of computations in: Factorization method.

11. Why factorization method is preferred over other methods?
a) Less calculations
b) No need of back substitution
c) No solving of cofactors
d) No elimination of unknowns

Explanation: Factorization method is preferred over other methods because it involves less number of calculations.

12. Apply factorization method for solving the following equations.

3x + 2y + 7z = 4
2x + 3y + z = 5
3x + 4y + z = 7

a) z = -1/3, y = 8/3, x = 7/8
b) z = -5/3, y = 4/3, x = 7/8
c) z = 7/3, y = 8/3, x = 9/8
d) z = -1/3, y = 1/3, x = 5/8

Explanation: Here,
Let $$\begin{bmatrix}1&0&0\\a&1&0\\b&c&1\end{bmatrix}$$$$\begin{bmatrix}p&q&r\\0&s&t\\0&0&u\end{bmatrix}$$ = $$\begin{bmatrix}3&2&7\\2&3&1\\3&4&1\end{bmatrix}$$
p = 3, q = 2, r = 7
ap = 2, aq + s = 3, ar + t = 1
bp = 3, bq + cs = 4, br + ct + u = 1
By solving these equations, we get
a = 2/3
b = 1
s = 5/3
c = 6/5
t = -11/3
u = -8/5
Thus, A =$$\begin{bmatrix}1&0&0\\2/3&1&0\\1&6/5&1\end{bmatrix}$$$$\begin{bmatrix}3&2&7\\0&5/3&-11/3\\0&0&-8/5\end{bmatrix}$$
.
Writing UX = V, the given system becomes
$$\begin{bmatrix}1&0&0\\2/3&1&0\\1&6/5&1\end{bmatrix}$$$$\begin{bmatrix}i\\j\\k\end{bmatrix}$$ = $$\begin{bmatrix}4\\5\\7\end{bmatrix}$$
Solving these equations, we have i = 4
(2/3)i + j = 5 or j = 7/3
i + (6/5)j + k = 7 or k = 1/5
Hence, the original system becomes,
$$\begin{bmatrix}3&2&7\\0&5/3&-11/3\\0&0&-6/5\end{bmatrix}$$$$\begin{bmatrix}4\\5\\7\end{bmatrix}$$ = $$\begin{bmatrix}4\\7/3\\1/5\end{bmatrix}$$
3x + 2y + 7z = 4
(5/3y) – 11/3z = 7/3
-8/5z = 1/5
By back substitution, we have
z = -1/3, y = 8/3, x = 7/8.

13. Apply factorization method for solving the following equations.

2x + 3y + z = 9
x + 2y + 3z = 6
3x + y + 2z = 8

a) z = 1/2, y = 1/2, x = 7/2
b) z = 5/2, y = 1/2, x = 9/2
c) z = 7/2, y = 3/2, x = 3/2
d) z = 1/2, y = 5/2, x = 3/2

Explanation: Here,
Let $$\begin{bmatrix}1&0&0\\a&1&0\\b&c&1\end{bmatrix}$$$$\begin{bmatrix}p&q&r\\0&s&t\\0&0&u\end{bmatrix}$$ = $$\begin{bmatrix}2&3&1\\1&2&3\\3&1&2\end{bmatrix}$$
p = 2, q = 3, r = 1
ap = 1, aq + s = 2, ar + t = 3
bp = 3, bq + cs = 1, br + ct + u = 2
By solving these equations, we get
a = 1/2
b = 3/2
s = 1/2
t = 5/2
c = -7
u = 18
Thus, A = $$\begin{bmatrix}1&0&0\\1/2&1&0\\3/2&-7&1\end{bmatrix}$$$$\begin{bmatrix}2&3&1\\0&1/2&5/2\\0&0&18\end{bmatrix}$$

Writing UX = V, the given system of equations become
$$\begin{bmatrix}1&0&0\\1/2&1&0\\3/2&-7&1\end{bmatrix}$$$$\begin{bmatrix}i\\j\\k\end{bmatrix}$$ = $$\begin{bmatrix}9\\6\\8\end{bmatrix}$$

i = 9
(1/2) i + j = 6
(3/2) i – 7j + k = 8
By back substitution,
i = 9, j = 3/2, k = 5
Hence, the original system becomes
$$\begin{bmatrix}2&3&1\\0&1/2&5/2\\0&0&18\end{bmatrix}$$ $$\begin{bmatrix}x\\y\\z\end{bmatrix}$$ = $$\begin{bmatrix}9\\3/2\\5\end{bmatrix}$$

2x + 3y + z = 9
(1/2) y + (5/2) z = 3/2
10z = 5
z = 1/2
By back substitution,
z = 1/2, y = 1/2, x = 7/2.

14. While solving a system of linear equations, which of the following will be the order of the decomposed matrices; L and U?
a) Order of L = 3X1, Order of U = 1X3
b) Order of L = 3X2, Order of U = 2X3
c) Order of L = 3X3, Order of U = 3X3
d) Order of L = 3X4, Order of U = 4X3