If B, C are square matrices of order n and if A=B+C, BC=CB, C2=0, then for any positive integer p, Ap+1. - Sri Chaitanya Infinity Learn Best Online Courses for NCERT Solutions, CBSE, ICSE, JEE, NEET, Olympiad and Class 6 to 12

A
$B^{p} [B + (p + 1) C]$
B
$B^{p} [B + (p - 1) C]$
C
$B^{p} [B - (p - 1) C]$
D
$B^{n} [B + (p + 1) C]$

Solution:

$∴ A = B + C \Rightarrow A^{p + 1} = {(B + C)}^{p + 1} = {}^{p + 1}C_{0} B^{p + 1} + {}^{p + 1}C_{1} B^{p} C + {}^{p + 1}C_{2} B^{p - 1} C^{2} + . . . + {}^{p + 1}C_{p + 1} C^{p + 1} = B^{p + 1} + {}^{p + 1}C_{1} B^{p} C + 0 + 0 + . . . [∵ C^{2} = 0 \Rightarrow C^{2} = C^{3} = . . . = 0] = B^{p} [B + (p + 1) C]$

Hence, $A^{p + 1} = B^{p} [B + (p + 1) C]$

If $B, C$ are square matrices of order $n$ and if $A = B + C, B C = C B, C^{2} = 0$ , then for any positive integer $p$ , $A^{p + 1}$ .

Solution:

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