Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
840 views
in Algebra by (238k points)
closed by

The value of k, for which the following system of linear equations has a non-trivial solution.

x + 2y - 3z = 0

2x + y + z = 0

x - y + kz = 0


1. 4
2. 2
3. 3
4. -4

1 Answer

0 votes
by (240k points)
selected by
 
Best answer
Correct Answer - Option 1 : 4

Concept:

Consider the system of m linear equations

a11 x1 + a12 x2 + … + a1n xn = 0

a21 x1 + a22 x2 + … + a2n xn = 0

am1 x1 + am2 x2 + … + amn xn = 0

The above equations containing the n unknowns x1, x2, …, xn. To determine whether the above system of equations is consistent or not, we need to find the rank of the following matrix.

\(A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \ldots &{{a_{1n}}}\\ {{a_{21}}}&{{a_{22}}}& \ldots &{{a_{2n}}}\\ \ldots & \ldots & \ldots & \ldots \\ {{a_{m1}}}&{{a_{m2}}}& \ldots &{{a_{mn}}} \end{array}} \right]\)

A is the coefficient matrix of the given system of equations.

  • The system of homogeneous equations has a unique solution (trivial solution) if and only if the determinant of A is non-zero.
  • The system of homogeneous equations has a Non - trivial solution if and only if the determinant of A is zero.

Calculation:

Given:

x + 2y – 3z = 0

2x + y + z = 0

x – y + kz = 0

\(\left[ {\begin{array}{*{20}{c}} 1&2&{ - 3}\\ 2&1&1\\ 1&{ - 1}&k \end{array}} \right]\)

For non-trivial solution the determinant should be zero

∴ 1(k + 1) – 2(2k - 1) – 3(-2 - 1) = 0

∴ k + 1 – 4k + 2 + 9 = 0

∴ 12 = 3k

∴ k = 4

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

...