Question

If $B,C$ are square matrices of order $n$ if $A=B+C,BC=CB,C^2=0,$ then for any positive integer $p,A^{p+1}=B^k[B+(p+1\left)C\right],$ where $k=$

• A. $p$
• B. $p+1$
• C. $p+2$
• D. $p-1$

Answer 1

The correct option is A $p$

$A^{p+1}={(B+C)}^{p+1}$

${=}^{p+1}{c}_0{B}^{p+1}{+}^{p+1}{c}_1{B}^{p}C{+}^{p+1}{c}_2{B}^{p-1}{C}^2+....{+}^{p+1}{c}_r{B}^{p+1-r}{C}^r$ (Using $BC=CB$)

But $C^2=0,C^3=C^4=.....=C^r=0$

Thus $A^{p+1}{=}^{p+1}{c}_0{B}^{p+1}{+}^{p+1}{c}_1{B}^{p}C=B^p[C(1+p)+B]$

$\Rightarrow k=p$