any help ful help would be appreciated.. each of the followings give an example
ID: 3185440 • Letter: A
Question
any help ful help would be appreciated.. each of the followings give an example of two equations with two unknowns, where the system will ve exactly one solution, no solution and infinite solutions. Explain the condition for each case based on the rank of A, the rank of A and the number of unknowns 5 A homogenous systern of linear equations always has the trivial solution “ zero solution" , give an example of a system of equations that has 2 equations with two unknowns such that (a) The trivial solution is the only answer (b) A non-trivial solution exist. Explain your ansrwver based on the rank of A compared to the number of unknowns and the number of independent equations.Explanation / Answer
4. (a). Let the 2 equations be x + y = 4…(1) and 2x+3y = 10…(2). Then the augmented matrix of this linear system is A =
1
1
4
2
3
10
The RREF of A is
1
0
2
0
1
2
Thus, the rank(A) = 2, and the rank of the coefficient matrix is also 2, which are equal to the no. of unknowns so that the above linear system has a unique solution.
(b). Let the 2 equations be x + y = 4…(1) and 2x+2y = 6…(2). Then the augmented matrix of this linear system is A =
1
1
4
2
2
6
The RREF of A is
1
0
0
0
0
1
Here, rank (A) = 2, and the rank of the coefficient matrix is 1 so that there is no solution to the above linear system .
(c ). Let the 2 equations be x + y = 4…(1) and 2x+2y = 6…(2). Then the augmented matrix of this linear system is A =
1
1
4
2
2
8
The RREF of A is
1
1
4
0
0
0
Here, the rank(A) = 1,and the rank of the coefficient matrix is also 1 which is less than the no. of unknowns, so that the above linear system has infinite solutions.
5. (a). Let the 2 equations be 3x + 5y = 0…(1) and 5x+3y = 0…(2). Then the coefficient matrix of this linear system is A =
3
5
5
3
The RREF of A is
1
0
0
1
The rank of A is 2, which is equal to the no. of unknowns so that the above linear system has only the trivial solution.
(b). Let the 2 equations be x + 2y = 0…(1) and 2x+4y = 0…(2). Then the coefficient matrix of this linear system is A =
1
2
2
4
The RREF of A is
1
2
0
0
The rank of A is 1, which is less than the no. of unknowns so that the above linear system has non-trivial solutions also.
1
1
4
2
3
10
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.