Hello, stephy7878!

Given the transition matrix: .

Find the fixed probability vector.

We want a row vector: .

We have: .

Multiply: .

The equations simplify to: .

Add [1] and [2]: . . . . whch is equivalent to [3].

So we need another equaton.

Here it is! . . . . . . . This is true forfixed probability vectors.all

From [3], we have: .

Substitute into [2]: .

Substitute into [4]: .

. . . Then: .

. . And: .

Therefore, the fixed probabiity vector is: .