MCQ
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 $\ce{(A + B)^2 = A^2 + B^2 + 2AB}$.
Reason $(R):$ For the matrices $A$ and $B$ given in assertion, $\text{AB=BA}$.
  • Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$
  • B
    Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A).$
  • C
    $(A)$ is true but $(R)$ is false.
  • D
    $(A)$ is false but $(R)$ is true.

Answer

Correct option: A.
Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$
$\text{AB=AI=A}$ and $\text{BA=IA=A}$
$\Rightarrow \text{AB=BA}$
Consequently, $\ce{(A + B)^2=(A + B)(A + B)}$
$=\ce{A(A + B) + B(A + B)=A^2 + AB + BA + B^2}$
$=\ce{A^2 + AB + AB + B^2=A^2 + 2AB + B^2}$

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