Question

Let $A=\left[a_{ij}\right]$ be $3\times 3$ matrix given by
$a_{ij}=\left\{\begin{array}{cc}\left(\frac{i+j}{2}\right)+\frac{|i-j|}{2},& ifi\ne j\\ \frac{i^j-(i.j)}{i^2+j^2},& ifi=j\end{array}\right\}$
where $a_{ij}$ denotes element of $i^{th}$ row & $j^{th}$ column of matrix A.If $A=p{A}^2-q{A}^{-1}-rI$, then remainder when $(p+q+r{)}^{11}$ is divided by 7 is

Answer 1

$A=\left[\begin{array}{ccc}0& 2& 3\\ 2& 0& 3\\ 3& 3& 1\end{array}\right]$
$|A|=32\Rightarrow A^{-1}$ will exist.
Characteristic equation of matrix A is
$|A-\lambda I|=0$
${\lambda }^3-{\lambda }^2-22\lambda -32=0$
So $A^3-A^2-22A=32I$
$A^2-A-22I=32A^{-1}$
$A=A^2-32A^{-1}-22I$
So p+q + r =55
${55}^{11}=(56-1{)}^{11}=(7K-1)$
So remainder is 6