Prove by method of induction((3, -4), (1, -1))^n = ((2n + 1, -4n), (n, -2n+1)), ∀ n ∈ N.

Question

Prove by method of induction

\(\begin{bmatrix}3&-4\\1&-1\end{bmatrix}^n\)\(=\begin{bmatrix}2n+1&-4n\\n&-2n+1 \end{bmatrix},\forall n\in N\) 

((3, -4), (1, -1))^n = ((2n + 1, -4n),  (n, -2n+1)), ∀ n ∈ N.

Answer 1

Let P(n) =  \(\begin{bmatrix}3&-4\\1&-1\end{bmatrix}^n\)\(=\begin{bmatrix}2n+1&-4n\\n&-2n+1 \end{bmatrix}\), for all n ∈ N.

Step I:

Step IV:

From all the steps above, by the principle of mathematical induction, P(n) is true for all n ∈ N.

 \(\begin{bmatrix}3&-4\\1&-1\end{bmatrix}^n\)\(=\begin{bmatrix}2n+1&-4n\\n&-2n+1 \end{bmatrix}\), \(\forall\) n ∈ N.