If B is a 3×3 matrix such that B^2=0, then det. I+B^50-50B is equal to :

Correct answer: (A) 1. B^2=0⇒B^4=B^6=B^8=....=B^50=0 and B^3=B^2B=0B=0 ⇒ I+B^50=I+C150B (higher powers are zero) ∴ det.I+B^50-50B =det. I+^50C_1B-50B =det.I =1

Q43 / 90 · JEE Main 2014 (09 Apr Online) · Mathematics

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

Correct answer
A
$1$
B
$2$
C
$3$
D
$50$

Solution

B^{2} = 0 \Rightarrow B^{4} = B^{6} = B^{8} = . . . . = B^{50} = 0

and

B^{3} = B^{2} B = 0 B = 0

\Rightarrow {(I + B)}^{50} = I + {}^{50}C_{1} B

(higher powers are zero)

∴

d e t . [{(I + B)}^{50} - 50 B]

= d e t . [I +^{50} C_{1} B - 50 B]

= d e t . [I]

= 1

Hence, the correct answer is option: 'A'.

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If B is a 3×3 matrix such that B2=0, then det. I+B50-50B is equal to :

Solution

Relevant Questions

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