Question

If A and B are two matrices such that $AB=B$ and $BA=A$ and $(A^2+B^2)=\lambda (A+B)$.Considering $f\left(x\right)=|\left[\sin x\right]+\left[\cos x\right]|$; where [ ] is greatest integer function find $f\left(4\lambda \right)$

Answer 1

$AB=B$ and $BA=A$ ..... $\left(i\right)$
$\Rightarrow \left(BA\right)B=AB=B$ and $\left(AB\right)A=BA=A$
$\Rightarrow B\left(AB\right)=B$ and $A\left(BA\right)=A$ ....... [Associativity of matrices]
$\Rightarrow B^2=B$ and $A^2=A$
$\Rightarrow (A^2+B^2)=(A+B)$
Hence $\lambda =1$
$f\left(4\right)=|\left[\sin 4\right]+\left[\cos 4\right]|=2$ ....... $[\because |\sin x|\le 1,|\cos x|\le 1]$