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3 and 4 .Determinants and Matrices
hard
જો $B$ એ $3 \times 3$ શ્રેણિક છે કે જેથી $B^2 = 0$, તો $|( I+ B)^{50} -50B|$ = . . .
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
