If B is a 3 × 3 matrix such that B^2 = 0 , then det. [( I+ B)^{50} -50B] is equal to

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3 and 4 .Determinants and Matrices

hard

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$, then det. $[( I+ B)^{50} -50B]$ is equal to

A

$1$

B

$2$

C

$3$

D

$50$

(JEE MAIN-2014)

Solution

det $\left[ {{{\left( {I + B} \right)}^{50}} – 50B} \right]$

$=$ det $[ {\,^{50}}{C_0}I + {\,^{50}}{C_1}B + {\,^{50}}{C_2}{B^2} + {\,^{50}}{C_3}{B^3} + … +$ $ {\,^{50}}{C_{50}}{B^{50}}{B^{50}} – 50B]$

{All terms having ${B^n},2 \le n \le 50$

will be zero because given that ${B^2} = 0$ }

$=$ det $\left[ {I + 50B – 50B} \right]$$=$ det $[I]=1$

Standard 12

Mathematics

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(JEE MAIN-2020)