Question

If $A=\left[\begin{array}{cc}2& 0\\ -1& 1\end{array}\right]$, then find $A^n$ and give a proof of $A^n$ by mathematical induction.

• A. $A^n=\left[\begin{array}{cc}{2}^{n-1}& 0\\ 2^n& 1\end{array}\right]$
• B. $A^n=\left[\begin{array}{cc}{2}^n& 0\\ 1-2^n& 1\end{array}\right]$
• C. $A^n=\left[\begin{array}{cc}{2}^n& 0\\ -2^n& 1\end{array}\right]$
• D. $A^n=\left[\begin{array}{cc}{2}^{n+1}& 0\\ 1-2^n& 2\end{array}\right]$

Answer 1

Answer: (B)

$A^n=\left[\begin{array}{cc}{2}^n& 0\\ 1-2^n& 1\end{array}\right]$

$A=\left[\begin{array}{cc}2& 0\\ -1& 1\end{array}\right]$

Verifying options $n=1$

Option A

$A=\left[\begin{array}{cc}0& 0\\ 2& 1\end{array}\right]\to$not true

Option B

$A=\left[\begin{array}{cc}2& 1\\ -1& 1\end{array}\right]\to$true

Option C

$A=\left[\begin{array}{cc}2& 0\\ -2& 1\end{array}\right]\to$not true

Option D

$A=\left[\begin{array}{cc}4& 0\\ -1& 2\end{array}\right]\to$not true

Option B is correct