1. ## Matrix

Prove that if $\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}$ then $(\mathbf{A}+\mathbf{B})^n=\binom n0\mathbf{A}^n+\binom n1\mathbf{A}^{n-1}\mathbf{B}+\dots + \binom n{n-1} \mathbf{A}\mathbf{B}^{n-1}+\binom nn\mathbf{B}^n$.

2. Originally Posted by james_bond
Prove that if $\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}$ then $(\mathbf{A}+\mathbf{B})^n=\binom n0\mathbf{A}^n+\binom n1\mathbf{A}^{n-1}\mathbf{B}+\dots + \binom n{n-1} \mathbf{A}\mathbf{B}^{n-1}+\binom nn\mathbf{B}^n$.
Hi

You can first prove by induction that
$\mathbf{B}^k\mathbf{A}=\mathbf{A}\mathbf{B}^k$

And use this to prove by induction
$\left(\mathbf{A}+\mathbf{B}\right)^n=\sum_{k=0}^{n } \binom nk\mathbf{A}^k\mathbf{B}^{n-k}$

$\left(\mathbf{A}+\mathbf{B}\right)^{n+1}=\left(\su m_{k=0}^{n} \binom nk\mathbf{A}^k\mathbf{B}^{n-k}\right)\left(\mathbf{A}+\mathbf{B}\right)$

$\left(\mathbf{A}+\mathbf{B}\right)^{n+1}=\sum_{k=0 }^{n} \binom nk\mathbf{A}^k\mathbf{B}^{n-k}\mathbf{A}+\sum_{k=0}^{n} \binom nk\mathbf{A}^k\mathbf{B}^{n+1-k}$

$\left(\mathbf{A}+\mathbf{B}\right)^{n+1}=\sum_{k=0 }^{n} \binom nk\mathbf{A}^{k+1}\mathbf{B}^{n-k}+\sum_{k=0}^{n} \binom nk\mathbf{A}^k\mathbf{B}^{n+1-k}$