1. Tardigrade
  2. Question
  3. Mathematics
  4. Let B, C be n × n matrices such that A=B+C, B C=C B and C 2 is a null matrix. Then B2020[B+(2021) C]=

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

TS EAMCET 2020

Solution:

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