MCQ
$A$ and $B$ be $3 \times 3$ matrices such that $AB + A + B = 0$ , then
- ✓$(A + B)^2 = A^2 + 2AB + B^2$
- B$|A| = |B|$
- C$A^2 = B^2$
- DNone
$\Rightarrow \mathrm{A}(\mathrm{B}+\mathrm{I})+\mathrm{I}(\mathrm{B}+\mathrm{I}) \Rightarrow(\mathrm{A}+\mathrm{I})(\mathrm{B}+\mathrm{I})=\mathrm{I}$
so $(\mathrm{B}+\mathrm{I}) $ and $(\mathrm{A}+\mathrm{I})$ are inverse of each other
$({\text{B}} + {\text{I}})({\text{A}} + {\text{I}}) = ({\text{A}} + {\text{I}})({\text{B}} + {\text{I}}) \Rightarrow \boxed{{\text{AB}} = {\text{BA}}}$
so $(\mathrm{A}+\mathrm{B})^{2}=\mathrm{A}^{2}+2 \mathrm{AB}+\mathrm{B}^{2}$
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$f(x)=\left[\begin{array}{ll}{\left[e^{x}\right],} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,x<0 \\ a e^{x}+[x-1], \,\,\,\,\,\,\,\,\,0 \leq x<1 \\ b+[\sin (\pi x)], \,\,\,\,\,\,\,\,\,\,\,\,1 \leq x<2 \\ {\left[e^{-x}\right]-c,} \,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,x \geq 2\end{array}\right.$
where a,b,c $\in R$ and $[t]$ denotes greatest integer less than or equal to $t.$ Then, which of the following statements is true $?$