SATHEE: Matrices and Determinants 4 Question 7

Matrices and Determinants 4 Question 7

7. The set of all values of $λ$ for which the system of linear equations $x - 2 y - 2 z = λ x, x + 2 y + z = λ y$ and $- x - y = λ z$

has a non-trivial solution

(2019 Main, 12 Jan II)

(a) contains exactly two elements.

(b) contains more than two elements.

(d) is an empty set.

Show Answer

Answer:

Correct Answer: 7. (c)

Solution:

The given system of linear equations is

$\begin{aligned} x - 2 y - 2 z & = λ x \\ x + 2 y + z & = λ y \\ - x - y - λ z & = 0, \end{aligned}$

which can be rewritten as

$\begin{aligned} (1 - λ) x - 2 y - 2 z & = 0 \\ x + (2 - λ) y + z & = 0 \\ x + y + λ z & = 0 \end{aligned}$

Now, for non-trivial solution, we should have

$\begin{aligned} | \begin{array}{ccc} 1 - λ & - 2 & - 2 \\ 1 & 2 - λ & 1 \\ 1 & 1 & λ \end{array} | = 0 \\ [∵ If a_{1} x + b_{1} y + c_{1} z = 0; a_{2} x + b_{2} y + c_{2} z = 0 \\ a_{3} x + b_{3} y + c_{3} z = 0] \\ has a non-trivial solution, then | \begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} | = 0 \\ \Rightarrow (1 - λ) [(2 - λ) λ - 1] + 2 [λ - 1] - 2 [1 - 2 + λ] = 0 \\ \Rightarrow (λ - 1) [λ^{2} - 2 λ + 1 + 2 - 2] = 0 \\ \Rightarrow (λ - 1)^{3} = 0 \\ \Rightarrow λ = 1 \end{aligned}$

Matrices and Determinants 4 Question 7

7. The set of all values of $λ$ for which the system of linear equations $x - 2 y - 2 z = λ x, x + 2 y + z = λ y$ and $- x - y = λ z$

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Matrices and Determinants 4 Question 7

7. The set of all values of λ for which the system of linear equations x−2y−2z=λx,x+2y+z=λy and −x−y=λz

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

7. The set of all values of $λ$ for which the system of linear equations $x - 2 y - 2 z = λ x, x + 2 y + z = λ y$ and $- x - y = λ z$