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Q.
The number of integral value(s) of $k$ such that the system of equations $kx-2y-z=x, \, ky-z=z+3x$ and $2x+kz=2y-z$ has non-trivial solution, is/are
NTA AbhyasNTA Abhyas 2020Matrices
Solution:
Rearranging all the equations, we get,
$\left(k - 1\right)x-2y-z=0$
$3x-ky+2z=0$
$2x-2y+\left(k + 1\right)z=0$
For a non-trivial solution,
$\begin{vmatrix} \left(k - 1\right) & -2 & -1 \\ 3 & -k & 2 \\ 2 & -2 & k+1 \end{vmatrix}=0\Rightarrow \left(k - 1\right)\left[- k^{2} - k + 4\right]+2\left[3 k - 1\right]-1\left[- 6 + 2 k\right]=0$
$\Rightarrow -k^{3}-k^{2}+4k+k^{2}+k-4+6k-2+6-2k=0$
$\Rightarrow -k^{3}+9k=0\Rightarrow k=0,-3,3$