Mathematics GATE EE

Question

The maximum value of a such that the matrix below has three linearly independent real eigen vectors is

$\begin{pmatrix} -3& 0 &-2 \\ 1& -1 & 0\\ 0& a & 2 \end{pmatrix}$

(a) $\frac{2}{3\sqrt{3}}$

(b) $\frac{1}{3\sqrt{3}}$

(c) $\frac{1+2\sqrt{3}}{3\sqrt{3}}$

(d)$\frac{1+\sqrt{3}}{3\sqrt{3}}$

Comments

@ srestha  MAM I AM GETTING (-27-10√3)/9   .....

 

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Starting solution.

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can someone explain in English how to solve this? Given solution is too big and not clear

Answer 1

Answer is option b)

Comments

I forgot to add the starting of the solution ,here it is

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There is no use of limiting Characteristic polynomial at Maxima point, and this step is redundant.

Since we need the maximum value of ‘a’ and we see in our polynomial that greater the value of ‘a’,  more the graph(polynomial) would shift above.

Hence we need to apply the boundary condition just at the minima point so that it doesn’t go above the x-axis and thus giving 3 distinct solution.