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Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMatrices1 Mark
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 $(A+B)^2=A^2+B^2+2 A B$.
Reason (R): For the matrices $A$ and $B$ given in assertion, $A B=B A$.
  • A
    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

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

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