Advertisements
Advertisements
Question
Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have infinitely many solution
Solution
Martix form `[("k", -2, 1),(1, -2"k", 1),(1, -2, "k")][(x),(y),(z)] = [(1),(-2),(1)]`
AX = B
Augmented martix [A|B] = `[("k", -2, 1, |, 1),(1, -2"k", 1, |, -2),(1, -2, "k", |, 1)]`
`{:("R"_1 ↔ "R"_3),(->):} [(1, -2, "k", |, 1),(1, -2"k", 1, |, -2),("k", -2, 1, |, 1)]`
`{:("R"_3 -> "R"_3 + "R"_2), (->):} [(1, -2, "k", |, 1),(0, -2"k" + 2, 1 - "k", |, -3),(0, -2 + 2"k", 1 - "k"^2, |, 1 - "k")]`
`{:("R"_3 -> "R"_3 + "R"_2),(->):} [(1, -2, "k", |, 1),(0, 2(1 - "k"), (1 - "k"), |, -3),(0, 0, 2 - "k" - "k"^2, |, ("k" + 2))]`
`-> [(1, -2, "k", |, 1),(0, 2(1 - "k"), (1 - "k"), |, -3),(0, 0, ("k" + 2)(1 - "k"), |, ("k" + 2))]`
∵ 2 – k – k2 = – (k2 + k – 2)
= – (k + 2)(k – 1)
= (k + 2)(1 – k)
Case:
If k = – 2
ρ(A) = 2, ρ(A|B) = 2
The system is consistent and it has infinitely many solution.
APPEARS IN
RELATED QUESTIONS
Test for consistency and if possible, solve the following systems of equations by rank method:
x – y + 2z = 2, 2x + y + 4z = 7, 4x – y + z = 4
Test for consistency and if possible, solve the following systems of equations by rank method:
3x + y + z = 2, x – 3y + 2z = 1, 7x – y + 4z = 5
Test for consistency and if possible, solve the following systems of equations by rank method:
2x – y + z = 2, 6x – 3y + 3z = 6, 4x – 2y + 2z = 4
Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have no solution
Find the value of k for which the equations kx – 2y + z = 1, x – 2ky + z = -2, x – 2y + kz = 1 have unique solution
Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have no solution
Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have a unique solution
Investigate the values of λ and µ the system of linear equations 2x + 3y + 5z = 9, 7x + 3y – 5z = 8, 2x + 3y + λz = µ, have an infinite number of solutions
Solve the following system of homogenous equations:
3x + 2y + 7z = 0, 4x – 3y – 2z = 0, 5x + 9y + 23z = 0
Solve the following system of homogenous equations:
2x + 3y – z = 0, x – y – 2z = 0, 3x + y + 3z = 0
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 a unique solution
By using Gaussian elimination method, balance the chemical reaction equation:
\[\ce{C2H + O2 -> H2O + CO2}\]
