Question

$A,B,C$ are three square matrices of order $n$ such that $A=B+C$ and $C$ is a nilpotent matrix with index $2$. If $B$ and $C$ commute, then $A^{10}$ is equal to

• A. $B^{10}+12B^{9}C$
• B. $B^{10}+10B^{9}C$
• C. $B^{10}+12B^{10}C$
• D. $B^{10}+10B^{10}C$

Answer 1

The correct option is B $B^{10}+10B^{9}C$

$C$ is nilpotent matrix with index $2.$
$\Rightarrow C^2=0$
$B$ and $C$ commute.
$\Rightarrow BC=CB$

$A=B+C{A}^2=(B+C)(B+C)=B^2+BC+CB+C^2=B^2+2BC$

$A^3=(B^2+2BC)(B+C)=B^3+B^{2}C+2BCB+2B{C}^2=B^3+B^{2}C+2B^{2}C+O=B^3+3B^{2}C$

$A^4=B^4+4B^{3}C$
$⋮$
$A^n=B^n+n{B}^{n-1}C\Rightarrow A^{10}=B^{10}+10B^{9}C$