Assertion (A) : A= [ ll 2 & 3 1 & 4 ], B= [ ll 1 & 0 0 & 1 ], the… - Vidyadip

Question

Assertion $(A) : A=\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right], B=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, then $(A+B)^2=A^2+B^2+2 A B$.
Reason $(R)$: For the matrices $A$ and $B$ given in assertion, $A B=B A$.

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Answer

$ A=\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right] $ and $B=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=I $
$ A B=A I=A $ and $ B A=I A=A $
$ \Rightarrow A B=B A$
Consequently, $(A+B)^2=(A+B)(A+B) $
$=A(A+B)+B(A+B)=A^2+A B+B A+B^2 $
$=A^2+A B+A B+B^2=A^2+2 A B+B^2$

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