arthham

JEE Advanced 2011 · Determinants

Let M and N be two 3×33 \times 3 non-singular skew- symmetric matrices such that MN=NMMN = NM. If PTP^{T} denotes the transpose of P, then M2N2(MTN)−1(MN−1)TM^{2}N^{2}(M^{T}N)^{-1}(MN^{-1})^{T} is equal to

Answers: C. −M2-M^{2}

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Let M and N be two 3 × 3 non-singular skew- symmetric matrices such that MN = NM. If P^{T} denotes the transpose of P, then M²N²(M^{T}N)⁻¹(MN⁻¹)^{T} is equal to

MT=−M,NT=−NM^{T} = -M, \qquad N^{T} = -N

MN=NM,M,N non-singularMN = NM, \qquad M, N \text{ non-singular}

M2N2(MTN)−1(MN−1)T=?M^{2}N^{2}(M^{T}N)^{-1}(MN^{-1})^{T} = ?

1. use transpose/inverse rules1.\ \text{use transpose/inverse rules}

2. simplify (MTN)−12.\ \text{simplify } (M^{T}N)^{-1}

3. simplify (MN−1)T, then combine3.\ \text{simplify } (MN^{-1})^{T}, \text{ then combine}

(AB)−1=B−1A−1,(AB)T=BTAT(AB)^{-1} = B^{-1}A^{-1}, \qquad (AB)^{T} = B^{T}A^{T}

(AT)−1=(A−1)T(A^{T})^{-1} = (A^{-1})^{T}

MTN=−MN ⇒ (MTN)−1=−(MN)−1M^{T}N = -MN \ \Rightarrow\ (M^{T}N)^{-1} = -(MN)^{-1}

=−N−1M−1= -N^{-1}M^{-1}

(MN−1)T=(N−1)TMT=(NT)−1MT(MN^{-1})^{T} = (N^{-1})^{T}M^{T} = (N^{T})^{-1}M^{T}

=(−N)−1(−M)=N−1M= (-N)^{-1}(-M) = N^{-1}M

expr=M2N2 (−N−1M−1) (N−1M)\text{expr} = M^{2}N^{2}\,(-N^{-1}M^{-1})\,(N^{-1}M)

=−M2N2N−1M−1N−1M= -M^{2}N^{2}N^{-1}M^{-1}N^{-1}M

=−M2 (M−1M) (N2N−1N−1)= -M^{2}\,(M^{-1}M)\,(N^{2}N^{-1}N^{-1})

=−M2⋅I⋅I=−M2= -M^{2}\cdot I\cdot I = -M^{2}

=−M2⇒option (3)= -M^{2} \Rightarrow \text{option (3)}

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