यदि A = [1 a 0 1 ] हो, तो A^4 बराबर है-

MCQ

यदि $A = \left[1\ a\ 0\ 1\right]$ हो$,$ तो $A^4$ बराबर है$-$

A
$\left[\begin{array}{cc}1 & a^4 \\ 0 & 1\end{array}\right]$
B
$\left[\begin{array}{cc}4 & 4 a \\ 0 & 4\end{array}\right]$
C
$\left[\begin{array}{cc}4 & a^4 \\ 0 & 4\end{array}\right]$
✓
$\left[\begin{array}{cc}1 & 4 a \\ 0 & 1\end{array}\right]$

✓

Answer

Correct option: D.

$\left[\begin{array}{cc}1 & 4 a \\ 0 & 1\end{array}\right]$

$ 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]$
$A ^4= A ^2 A^2=\left[\begin{array}{cc}1 & 2 a \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}1 & 2 a \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}1+0 & 2 a+2 a \\ 0+0 & 0+1\end{array}\right]$
$=\left[\begin{array}{cc}1 & 4 a \\ 0 & 1\end{array}\right]$
अतः सही विकल्प $(D)$ है।

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