The system of linear equations 

-y + z = 0

(4d - 1) x + y + Z = 0

(4d - 1) z = 0 

has a non-trivial solution, if d equals 

Concept:

For a homogeneous system of linear equations:

Having non-trivial solution:

The rank of the matrix should be less than the number of variables.

Or determinant of the matrix should be equal to zero.

Calculation:

Given:

-y + z = 0

(4d - 1) x + y + Z = 0

(4d - 1) z = 0

So, A = \(\begin{vmatrix} 0 &-1 &1 \\ (4d - 1) &~~1 &1 \\ 0 &~~0 &(4d - 1) \\ \end{vmatrix}\)

For non-trivial solution:

det. A = 0

⇒ \(\left| A\right|~=~0\)

⇒ 0 × [(4d - 1) - 0] + 1 × [(4d - 1)² - 0] + 1(0 - 0) = 0

⇒ (4d - 1)² = 0

⇒ d = \(\frac{1}{4}, \frac{1}{4}\)

∴ The system of linear equations has a non-trivial solution if d equals to \(\frac{1}{4}\)