École Centrale School of Engineering, Mahindra University, Hyderabad
ES211 (Numerical Methods), Problem Sheet−II
1. * Solve the following system of equations using (a) Gauss-elimination, (b) Doolittles’, and (c) Crouts’
decomposition
x1 + x2 + x3 = 1, 4x1 + 3x2 − x3 = 6, 3x1 + 5x2 + 3x3 = 4.
2. If A ∈ Rn×n is symmetric and non-singular, then prove that A1 = [a1i,j ], where a1i,j = ai,j −
ai,1
a1,j ,
a1,1
i, j = 1, 2, . . . , n is also symmetric.
3. Prove that inverse of a unit upper triangular matrix is a unit upper triangular matrix.
4. Let A ∈ Rn×n be a non-singular matrix which can be written as A = LU. Let Ak , k = 1, 2, . . . , n denotes the
principal leading sub-matrix of order k. Show that det(Ak ) 6= 0, k = 1, 2, . . . , n.
5. * Show that the following matrix A is symmetric and positive definite, then solve the system Ax = b where


12 4 −1




T

A= 4 7 1 
(1)
 , b = [15, 12, 6]


−1 1 6
6. * Solve the following system of equations using (a) Gauss-elimination with partial pivoting, and (b) Gausselimination with complete pivoting
6x1 − 2x2 + 2x3 + 4x4 = 16, 12x1 − 8x2 + 6x3 + 10x4 = 26,
3x1 − 13x2 + 9x3 + 3x4 = −19, −6x1 + 4x2 + x3 − 18x4 = −34.
7. * Discuss Thomas algorithm for solving a tri-diagonal system of equations. Then solve the following system
with n = 3.
4x1 − x2 = −20, xj−1 − 4xj + xj+1 = 40 (2 ≤ j ≤ n − 1), −xn−1 + 4xn = −20.
Practice problems for Lab
Write a MATLAB code for the problems marked with *
1
Home work (* Submit solutions of these problems as an assignment on or before March 12, 2021)
1. * Show that if A ∈ Rn×n is non-singular, and A x = b is altered by multiplication of its j th column by α 6= 0,
1
then the solution is altered only in the j th component, which is multiplied by .
α
2. Let L = [li,j ], i, j = 1, 2, . . . , n, M = [mi,j ], i, j = 1, 2, . . . , n be unit lower triangular matrices. Show that LM
is also a unit lower triangular matrix.
3. * Prove that the Thomas algorithm for solving a tri- diagonal system needs only O(n) floating point operations.
4. Prove that inverse of a unit lower triangular matrix is a unit lower triangular matrix.
5. * If A ∈ Rn×n is a symmetric and positive definite matrix ⇐⇒ A can be written as A = GGT , where G is
a lower triangular matrix.
6. * Find the (a) Doolittles’, and (b) Crouts’ decompositions of

60


A=
 30

20
30
20
15
20



15 


12
7. * Find the inverse of the following matrix using Cholesky decomposition.


4
−1
0
0






 −1 4 −1 0 


A=

 0 −1 4 −1 




0
0 −1 4
2