Question

If $A=\left[\begin{array}{cc}1& a\\ 0& 1\end{array}\right]$, then $A^n$ (where $nϵN$) equals.

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

Answer 1

The correct option is A $\left[\begin{array}{cc}1& na\\ 0& 1\end{array}\right]$

Given,

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

Now,

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

or, $A^2=\left[\begin{array}{cc}1& 2a\\ 0& 1\end{array}\right]$.

Again,

$A^3=\left[\begin{array}{cc}1& 2a\\ 0& 1\end{array}\right]$$\left[\begin{array}{cc}1& a\\ 0& 1\end{array}\right]$

or, $A^3=\left[\begin{array}{cc}1& 3a\\ 0& 1\end{array}\right]$.

Proceeding in this way using mathematical induction we've,

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