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The following questions consist of two statements, one labelled as the 'Assertion $(A)^{\prime}$ and the other as 'Reason $(R)^{\prime}$. You are to examine these two statements canefully and select the answer.
$Assertion$ $(\mathbf{A}):$ If $\mathrm{A}=\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right), \mathrm{B}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$, then $(\mathrm{A}+\mathrm{B})^{2}$
$=\mathrm{A}^{2}+\mathrm{B}^{2}+2 \mathrm{AB}$
$Reason(R)$: In the above $A B=B A$
Options:
$Assertion$ $(\mathbf{A}):$ If $\mathrm{A}=\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right), \mathrm{B}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$, then $(\mathrm{A}+\mathrm{B})^{2}$
$=\mathrm{A}^{2}+\mathrm{B}^{2}+2 \mathrm{AB}$
$Reason(R)$: In the above $A B=B A$
Solution:
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Verified Answer
The correct answer is:
Both A and $\mathrm{R}$ are individually true and $\mathrm{R}$ is the correct explanation of A
$A=\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right), B=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$
$\mathrm{AB}=\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$
$=\left(\begin{array}{ll}2+0 & 0+3 \\ 1+0 & 0+4\end{array}\right)=\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)$
$\mathrm{BA}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)$
$=\left(\begin{array}{ll}2+0 & 3+0 \\ 0+1 & 0+4\end{array}\right)=\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)$
Also, $A+B=\left(\begin{array}{ll}3 & 3 \\ 1 & 5\end{array}\right)$
$(\mathrm{A}+\mathrm{B})^{2}=\left(\begin{array}{cc}9+3 & 9+15 \\ 3+5 & 3+25\end{array}\right)=\left(\begin{array}{cc}12 & 24 \\ 8 & 28\end{array}\right)$
$\mathrm{A}^{2}=\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)=\left(\begin{array}{ll}4+3 & 6+12 \\ 2+4 & 3+16\end{array}\right)=\left(\begin{array}{ll}7 & 18 \\ 6 & 19\end{array}\right)$
$\mathrm{B}^{2}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$
$\mathrm{A}^{2}+\mathrm{B}^{2}+2 \mathrm{AB}$
$=\left(\begin{array}{ll}7 & 18 \\ 6 & 19\end{array}\right)+\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)+2\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)$
$=\left(\begin{array}{cc}12 & 24 \\ 8 & 28\end{array}\right)=(\mathrm{A}+\mathrm{B})^{2}$
So, Assertion $\mathrm{A}$ is correct $\mathrm{R}$ is $\mathrm{AB}=\mathrm{BA}$
Hence, $\mathrm{R}$ is correct. Since this leads from Assersion A, then both A and $R$ are individually true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
$\mathrm{AB}=\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$
$=\left(\begin{array}{ll}2+0 & 0+3 \\ 1+0 & 0+4\end{array}\right)=\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)$
$\mathrm{BA}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)$
$=\left(\begin{array}{ll}2+0 & 3+0 \\ 0+1 & 0+4\end{array}\right)=\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)$
Also, $A+B=\left(\begin{array}{ll}3 & 3 \\ 1 & 5\end{array}\right)$
$(\mathrm{A}+\mathrm{B})^{2}=\left(\begin{array}{cc}9+3 & 9+15 \\ 3+5 & 3+25\end{array}\right)=\left(\begin{array}{cc}12 & 24 \\ 8 & 28\end{array}\right)$
$\mathrm{A}^{2}=\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)=\left(\begin{array}{ll}4+3 & 6+12 \\ 2+4 & 3+16\end{array}\right)=\left(\begin{array}{ll}7 & 18 \\ 6 & 19\end{array}\right)$
$\mathrm{B}^{2}=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)$
$\mathrm{A}^{2}+\mathrm{B}^{2}+2 \mathrm{AB}$
$=\left(\begin{array}{ll}7 & 18 \\ 6 & 19\end{array}\right)+\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)+2\left(\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right)$
$=\left(\begin{array}{cc}12 & 24 \\ 8 & 28\end{array}\right)=(\mathrm{A}+\mathrm{B})^{2}$
So, Assertion $\mathrm{A}$ is correct $\mathrm{R}$ is $\mathrm{AB}=\mathrm{BA}$
Hence, $\mathrm{R}$ is correct. Since this leads from Assersion A, then both A and $R$ are individually true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$.
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