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