# Find all values of \lambda for which the homogeneous linear system (2\lambda +9) x -5y = 0...

## Question:

Find all values of {eq}\lambda {/eq} for which the homogeneous linear system

{eq}(2\lambda +9) x -5y = 0 \\ x - \lambda y = 0 {/eq}

has non trivial solutions.

## Homogneous Linear System:

The values of the parameter {eq}\lambda {/eq} for which the homogeneous linear system

{eq}(2\lambda +9) x -5y = 0 \\ x - \lambda y = 0 {/eq}

has non trivial solutions are calculated by setting to zero the determinant

of the coefficient matrix associated to the system, i.e.

{eq}\det (\mathbf A) = 0 \\ \mathbf A = \bigl(\begin{smallmatrix} 2\lambda +9 & -5\\ 1 & -\lambda \end{smallmatrix}\bigr) {/eq}

Upon setting to zero the determinant of the coefficient matrix associated to the system,

we find

{eq}\det (\mathbf A) = -\lambda(2\lambda +9) + 5 =0...

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