The system of equations x-k y-z=0, k x-y-z=0 and x+y-z=0 has...

The system of equations $x-k y-z=0, k x-y-z=0$ and $x+y-z=0$ has non trivial solutions. Then the possible values of $k$ are (a) $\pm 1$ (b) $\pm 2$ (c) 0 (d) $\pm 4$ (a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement- 1 (b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement- 1 (c) Statement- 1 is True, Statement- 2 is False (d) Statement- 1 is False, Statement- 2 is True

The system of equations $x-k y-z=0, k x-y-z=0$ and $x+y-z=0$ has non trivial solutions. Then the possible values of $k$ are
(a) $\pm 1$
(b) $\pm 2$
(c) 0
(d) $\pm 4$
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement- 1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement- 1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True

Key Concepts

Homogeneous Systems

A homogeneous system is a set of linear equations in which every constant term is zero. Such systems always have the trivial solution (all variables equal zero), but the focus is on identifying conditions under which nontrivial solutions exist, typically involving the coefficient matrix.

Nontrivial Solutions

Nontrivial solutions of a linear system refer to solutions other than the trivial zero solution. For a homogeneous system, these exist if and only if the determinant of the coefficient matrix is zero, which implies that the matrix is singular and does not have full rank.

Determinant Condition for Singular Matrices

The determinant condition is a crucial concept in linear algebra where a square matrix has a determinant equal to zero if and only if it is singular. For homogeneous systems, this condition is necessary and sufficient for the existence of nontrivial solutions, as it indicates that the rows (or columns) of the coefficient matrix are linearly dependent.

Parameter Dependence in Linear Systems

When a linear system involves a parameter within its coefficients, the system’s behavior and the existence of nontrivial solutions depend on the specific values of this parameter. Determining the parameter values that make the determinant of the coefficient matrix zero is essential for identifying when the system has nontrivial solutions.

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