Maximum value of a for 3 independent real eigen vectors.

Find the maximum value of a in the below matrix such that it has three linearly independent real eigen vectors.

$M= \left[ {\begin{array}{ccc} -3 & 0 & -2 \\ 1 & -1 & 0\\ 0 & a & -2\\ \end{array} } \right]$

In the solution, they found the characteristic equation of the matrix and maximized the value of $a$ using the normal procedure of Maxima and minima.

But I don't understand how does this prove that all the three eigen vectors of the matrix with that value of $a$ are linearly independent.

According to me, we need to find 3 distinct eigen values so that the eigen vectors of those eigen values are independent. To get 3 distinct values, rank has to be 3 and the condition should be $|A| \neq 0$. But I am not getting the right answer with this approach.

Can someone tell me why is the first approach working ?

• What do you mean by "maximized the value of $a$"? Maximized with respect to what? – rogerl Jan 6 '18 at 15:10
• @rogerl Maximum value of $a$ so that we get three linearly independent vectors . – Zephyr Jan 6 '18 at 15:59
• where do yo see the "solution"? – Siong Thye Goh Jan 6 '18 at 16:18
• @SiongThyeGoh Solution is long so I didn't post it. But the answer given is $a=\frac{1}{3√3}$. Can you just tell me the procedure please ? I don't want the exact answer. – Zephyr Jan 6 '18 at 16:46
• @Moo But how can you tell that the eigen vectors obtained from matrix M with that value of $a$ will be linearly independent? – Zephyr Jan 7 '18 at 4:33