Transition "political" matrix with an unkown, how to determine its value?

Question

The problem goes as follows: $$ P=\left( \begin{matrix} a & 0.6\\ 1-a & 0.4\\ \end{matrix} \right) $$

Determine the value of the parameter $a \in [0,1]$ for which $P$ does not have an inverse.

So then I know the value of $a$ lies between $0$ and $1$, inclusively. And since I don't get information on the current states of the transitional values, this has to be done algebraically.

Is that correct?

In that case, as it is a "political" transition matrix: L = left swing, R = right swing.

$$P \cdot\left(\begin{array}{c} x\\ y\end{array} \right) = \left( \begin{array}{c} L \\ R\end{array} \right)$$

$$ax + .6y = L \\x(1-a) + .4y = R$$

$$ax = L - .6y \\x(1-a)= R - .4y$$ $$a = \frac{L - .6y}{x} \\$$

Or am I lost?

Thanks beforehand for help.

Answer 1

Hint:

A matrix does not have an inverse when its determinant $=0$. The value of $a$ that does this is: $$a\times 0.4 - (1-a)\times 0.6 =0\implies a=\, ?$$

Answer 2

There is no inverse to a matrix with determinant 0. so when Det(P)=0, $P^{-1}$ doesn't exist.

i.e 0.4a-0.6(1-a)=0, 
so 0.4a+0.6a-0.6=0, 
so a=0.6