Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M ≠ N ^ 2 and M ^ 2 = N ^ 4 , then

Correct answer: A. Determinant of ( M ^ 2 + M N ^ 2 ) is 0 M ^ 2 = N ^ 4 ⇒ M ^ 2 - N ^ 4 = 0 ⇒ (M - N ^ 2 )(M + N ^ 2 ) = 0 (As M, N commute.) Also, M ≠ N ^ 2 , Det((M - N ^ 2 )(M + N ^ 2 )) = 0 As M - N ^ 2 is not null ⇒ Det(M + N ^ 2 ) = 0 Also Det( M ^ 2 + M N ^ 2 ) = (DetM)(Det(M + N ^ 2 )) = 0 ⇒ There exist non - null U such that ( M ^ 2 + M N ^ 2 )U = 0

JEE Advanced 2014Paper 1

Exam: JEE Advanced
Year: 2014
Paper: Paper 1
Subject: Mathematics
Chapter: Matrices

JEE Advanced 2014 · Paper 1 · Mathematics · Matrices

Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M...

Let M and N be two

3 \times 3

matrices such that MN = NM. Further, if

M \ne {N}^{2}

and

{M}^{2} = {N}^{4}

, then

Options

ADeterminant of $\left({M}^{2} + M{N}^{2}\right)$ is 0
BThere is a $3 \times 3$ non - zero matrix $\text{U}$ such that $\left(M^{2} + MN^{2}\right)U$ is the zero matrix
CDeterminant of $\left({M}^{2} + M{N}^{2}\right) \ge 1$
DFor a $3 \times 3$ matrix $\text{U}$ , if $\left(M^{2} + MN^{2}\right)U$ equals the zero matrix then $\text{U}$ is the zero matrix

Correct answer

A. Determinant of $\left({M}^{2} + M{N}^{2}\right)$ is 0

Step-by-step solution

{M}^{2} = {N}^{4} ⇒ {M}^{2} - {N}^{4} = 0 ⇒ \left(M - {N}^{2}\right)\left(M + {N}^{2}\right) = 0

(As M, N commute.)
Also,

M \ne {N}^{2}, Det\left(\left(M - {N}^{2}\right)\left(M + {N}^{2}\right)\right) = 0

M - {N}^{2} is not null

⇒ Det\left(M + {N}^{2}\right) = 0

<br />Also

Det\left({M}^{2} + M{N}^{2}\right) = \left(DetM\right)\left(Det\left(M + {N}^{2}\right)\right) = 0

<br />

⇒

There exist non - null U such that \left({M}^{2} + M{N}^{2}\right)U = 0

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