Find Non Trivial Solution Homogeneous System
SOLVING EQUATIONS FOR TRIVIAL AND NON TRIVIAL SOLUTIONS
Question 1 :
Solve the following system of homogenous equations.
(i) 3x + 2y + 7z = 0, 4x − 3y − 2z = 0, 5x + 9y + 23z = 0
Solution :
rank of (A) is 2 and rank of (A, B) is 2 < 3.
3x + 2y + 7z = 0 -----(1)
-17y - 34z = 0 -----(2)
Let z = t
-17y = 34t
y = 34t/(-17) = -2t
By applying the value of z in (1), we get
3x + 2(-2t) + 7t = 0
3x - 4t + 7t = 0
3x = -3t
x = -t
Hence the solution is (-t, -2t, t)
(ii) 2x + 3y − z = 0, x − y − 2z = 0, 3x + y + 3z = 0
Solution :
Rank of A is 3 and rank of (A, B) is 3.
Since rank of A and rank of (A, B) are equal, it has trivial solution.
Question 2 :
Determine the values of λ for which the following system of equations x + y + 3z = 0, 4x + 3y + λz = 0, 2x + y + 2z = 0 has (i) a unique solution (ii) a non-trivial solution.
Solution :
(i) a unique solution
If λ ≠ 8, then rank of A and rank of (A, B) will be equal to 3.It will have unique solution.
(ii) a non-trivial solution.
If λ = 8, then rank of A and rank of (A, B) will be equal to 2.It will have non trivial solution.
Question 3 :
By using Gaussian elimination method, balance the chemical reaction equation :
C2 H6+ O2 -> H2O + CO2
Solution :
x1 C 2 H 6 + x2 O 2 -> x3 H 2 O + x4 CO 2 ----(1)
The number of carbon atoms on the left-hand side of (1) should be equal to the number of carbonatoms on the right-hand side of (1). So we get a linear homogenous equation
2x 1 = x4 (carbon)
2x1 - x4 = 0 ----(1)
6x1 = 2x3 (Hydrogen)
6x1 - 2x3 = 0 ----(2)
2x2 = 1 x 3+2x 4 (Oxygen)
2x2 -x3- 2x4 = 0 ----(3)
rank of A is 3 = rank of (A, B) = 3 < 4
Then the system is consistent and it has infinitely many solution.
2x1 + 0x2 + 0x3 - x4 = 0 ---(A)
2x2 - x3 - 2x4 = 0 ---(B)
-2x3 + 3x4 = 0 ---(C)
Let x4 = t
-2x 3 = -3t
x3 = 3t/2
By applying the value of x 3 in (B), we get
2x2 - (3t/2) - 2t = 0
2x2 = (3t/2) + 2t
2 x 2 = (7t/2)
x2 = (7t/4)
By applying the value of x4 in (A), we get
2x1 - t = 0
x1 = t/2
Let t = 4
x 1 = 2, x 2 = 7,x3 = 6, x4 = 4.
x1 C2 H6+ x2 O2 -> x3 H2O + x4 CO2
2 C2 H6+ 7 O2 -> 6 H2O + 4 CO2
Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.
If you have any feedback about our math content, please mail us :
v4formath@gmail.com
We always appreciate your feedback.
You can also visit the following web pages on different stuff in math.
WORD PROBLEMS
HCF and LCM word problems
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Algebra word problems
Word problems on trains
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on unit price
Word problems on unit rate
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Double facts word problems
Trigonometry word problems
Percentage word problems
Profit and loss word problems
Markup and markdown word problems
Decimal word problems
Word problems on fractions
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Time and work word problems
Word problems on sets and venn diagrams
Word problems on ages
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Profit and loss shortcuts
Percentage shortcuts
Times table shortcuts
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
Sum of all three four digit numbers formed using 0, 1, 2, 3
Sum of all three four digit numbers formed using 1, 2, 5, 6
Find Non Trivial Solution Homogeneous System
Source: https://www.onlinemath4all.com/solving-equations-for-trivial-and-nontrivial-solutions.html