Prepared by:
Engr. Gilmark P. Repulda
SAMPLE PROBLEMS III
Numerical Solutions to Civil Engineering Problems
1 | P a g e
CENUMES 313
Civil Engineering
Gaussian Elimination Method
In Exercises 1-2, solve the system of linear equations by each of the methods
shown.
a. Gaussian elimination with back-substitution
b. Gauss-Jordan elimination
c. Cramerโs Rule
1. 3๐ฅ1+ 3๐ฅ2+ 5๐ฅ3=1
3๐ฅ1+ 5๐ฅ2+ 9๐ฅ3=2
5๐ฅ1+ 9๐ฅ2โ17๐ฅ3=4 2. 2๐ฅ1+ ๐ฅ2+ 2๐ฅ3=6
โ๐ฅ1+ 2๐ฅ2โ 3๐ฅ3=0
3๐ฅ1+ 2๐ฅ2โ ๐ฅ3=6
In Exercises 3-4, find the adjoint of the matrix.
3. [0 1
โ2 1] 4. [1 โ1 1
0 1 2
0 0 โ1]
5. Use linear equations to find the parabola ๐ฆ=๐๐ฅ2+๐๐ฅ+๐ that passes through the
points (โ1,2),(0,1), and (2,6).
In Exercises 6 and 7, use the LU-factorization of the coefficient matrix to solve
the linear system.
6. ๐ฅ โฌ +๐ง =3
2๐ฅ +๐ฆ +2๐ง =7
3๐ฅ +2๐ฆ +6๐ง =8 7. 2๐ฅ1+๐ฅ2+๐ฅ3โ๐ฅ4=7
โฌ 3๐ฅ2+๐ฅ3โ๐ฅ4=โ3
โฌ โฌ โ2๐ฅ3โฌ =2
2๐ฅ1+๐ฅ2+๐ฅ3โ2๐ฅ4=8
7. Find values of the constants A, B, C, and D that make the following equation
an identity (i.e., true for all values of x).
3๐ฅ3+4๐ฅ2โ6๐ฅ
(๐ฅ2+2๐ฅ+2)(๐ฅ2โ1)=๐ด๐ฅ+๐ต
(๐ฅ2+2๐ฅ+2)+๐ถ
๐ฅโ1+๐ท
๐ฅ+1
[Hint: Obtain a common denominator on the right, and then equate corresponding
coefficients of the various powers of ๐ฅ in the two numerators. Students of
calculus will recognize this as a problem in partial fractions.]
8. Find the inverse of
[1
2(๐๐ฅ+๐โ๐ฅ)1
2(๐๐ฅโ๐โ๐ฅ)
1
2(๐๐ฅโ๐โ๐ฅ)1
2(๐๐ฅ+๐โ๐ฅ)]
