. Investigate the values of and such that the system of equations
x + y + z = 6
x + 2y + 3z = 10
x + 2y + z = , may have
(i) Unique solution
(i)
For the system
x + y + z = 6
x + 2y + 3z = 10
x + 2y + Hz = K
to have unique solution, the determinant on the left hand side, namely
|1 1 1|
|1 2 3| = Δ
|1 2 H|
must be non-zero.
Δ would be zero is when there are two identical rows or columns. This can happen for one value of H.
You will need to stipulate the condition on H such that identical rows do not appear in Δ.
Post if you need further help.
To investigate the values of λ and μ such that the system of equations has a unique solution, we need to assess the consistency and dependency of the equations.
First, let's write the system of equations in matrix form:
A * X = B
where A is the coefficient matrix, X is the unknowns matrix, and B is the constant matrix.
The coefficient matrix A can be formed as follows:
A = [[1, 1, 1],
[1, 2, 3],
[1, 2, λ]]
The constant matrix B can be written as:
B = [[6],
[10],
[μ]]
To have a unique solution, the coefficient matrix A must be non-singular, which means that its determinant must be non-zero.
So, let's calculate the determinant of A:
det(A) = λ - 2λ + 2 = 2 - λ
For the system to have a unique solution, det(A) must not be equal to zero. Therefore, we set det(A) ≠ 0 and solve for λ:
2 - λ ≠ 0
λ ≠ 2
Thus, any value of λ except 2 will lead to a unique solution for the system of equations.
Now, the next step is to find the values of μ such that the system of equations is consistent, which means there exists at least one solution.
To determine the consistency, we can use row reduction techniques to obtain the matrix in row-echelon form. If we encounter a row of zeros in the row-echelon form, we need to verify that the corresponding entry in the constant matrix B is also zero. If this condition is met, then the system of equations is consistent; otherwise, it is inconsistent.
Let's apply row reduction to find the row-echelon form:
RREF(A | B) = [[1, 0, -λ + 3],
[0, 1, λ - 1],
[0, 0, -λ + μ]]
From the row-echelon form, we can see that the system is consistent for any value of μ.
In summary:
(i) To have a unique solution, λ ≠ 2.
(ii) The system is consistent for any value of μ.