proof of each row of self product of transition matrix sums to 1

I am unable to proof that the sum of each row of self product of a transition matrix is 1...

Let A be a transition probability matrix which means that each row of A sums to 1, and let P=A*A.

I want the prove that P is a also a valid transition matrix,i.e each row of P sums to 1.

Please Help.

Regards.

Given two transition matrices A (m x p) and B (p x n), we want to prove that C = AB (m x n) is a transition matrix.

We know that C_ij = Σ_k A_ik B_kj.

What is the sum of the ith row of C?

Σ_j C_ij = Σ_j Σ_k A_ik B_kj = Σ_k Σ_j A_ik B_kj = Σ_k A_ik (Σ_j B_kj) = Σ_k A_ik = 1

Therefore C is a transition matrix.