If A = [ ll 1 & a 0 & 1 ] then A ^4 equals to : - Vidyadip

Question

If $A =\left[\begin{array}{ll}1 & a \\ 0 & 1\end{array}\right]$ then $A ^4$ equals to :

✓

Answer

(D)
$
\begin{aligned}
A^2 & =A \cdot A=\left[\begin{array}{ll}
1 & a \\
0 & 1
\end{array}\right]\left[\begin{array}{ll}
1 & a \\
0 & 1
\end{array}\right] \\
A^2 & =\left[\begin{array}{cc}
1 & 2 a \\
0 & 1
\end{array}\right] \\
A^4=A^2 A^2 & =\left[\begin{array}{cc}
1 & 2 a \\
0 & 1
\end{array}\right]\left[\begin{array}{cc}
1 & 2 a \\
0 & 1
\end{array}\right]=\left[\begin{array}{cc}
1+0 & 2 a+2 a \\
0+0 & 0+1
\end{array}\right] \\
& =\left[\begin{array}{cc}
1 & 4 a \\
0 & 1
\end{array}\right]
\end{aligned}
$
Hence correct option is (D).

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