Question

Let $B,C$ be $n\times n$ matrices such that $A=B+C,BC=CB$ and $C^2$ is a null matrix. Then $B^{2020}[B+(2021)C]=$

• A. $A^{2020}$
• B. null zero matrix of order $n\times n$
• C. $A^{2021}$
• D. $B^{2021}$

Answer 1

Answer: (C)

Given that,
$A=B+C\dots$ (i)
$BC=CB$ and $C^2=0$
From Eq. (i)
$A^{n+1}=(B+C{)}^{n+1}$
$\Rightarrow A^{n+1}={}^{n+1}{C}_0\cdot B^{n+1}+{}^{n+1}{C}_{1}C\cdot B^n+\dots$
But, $C^2=0,C^3=C^4\dots =C^r=0$
$\therefore A^{n+1}={}^{n+1}{C}_0{B}^{n+1}+{}^{n+1}{C}_1{B}^n\cdot C$
$=B^{n+1}+(n+1)B^{n}C$
$A^{n+1}=B^n[B+(n+1)C]$
$\therefore B^{2020}[B+(2020+1)C]=A^{2020+1}$
$\Rightarrow B^{2020}[B+(2021)C]=A^{2021}$