Question

If $A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right];I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then which one of the following holds for all $n\ge 1$?

• A. $A^n=2^{n-1}A+(n-1)I$
• B. $A^n=n4+(n-1)I$
• C. $A^n=2^{-1}A-(n-1)I$
• D. $A^n-nA-(n-1)I$

Answer 1

The correct option is C $A^n=2^{-1}A-(n-1)I$

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

(A) For $n=2,A^2=2A+I$ . First element $=2\times 1+1=3\ne 1$

(B) For $n=2,A^2=2A+I$ (not clear)

(C) $A^2=2A-I,a_{11}=1a_{12}=0a_{21}=2a_{22}=1$

$\Rightarrow$ option (C)