Let A= I 2 -2 MM T , where M is real matrix of order 2 × 1 such that the relation M T M = I 1 holds. If λ is a real number such that the relation A X=λ X holds for some non-zero real matrix X of order 2 × 1 , then the sum of squares of all possible values of λ is equal to :

Mathematics · Matrices

JEE Main 2024 — 1 February, Shift 2 — Question 22

Let $A=\mathrm{I}_{2}-2 \mathrm{MM}^{\mathrm{T}}$ , where M is real matrix of order $2 \times 1$ such that the relation $\mathrm{M}^{\mathrm{T}} \mathrm{M}=\mathrm{I}_{1}$ holds. If $\lambda$ is a real number such that the relation $A X=\lambda X$ holds for some non-zero real matrix X of order $2 \times 1$ , then the sum of squares of all possible values of $\lambda$ is equal to :

Answer: 2

Numerical answer — enter this value.

Step-by-step solution

$A=I_{2}-2 \mathrm{MM}^{\mathrm{T}}$

$A^{2}=\left(I_{2}-2 M M^{T}\right)\left(I_{2}-2 M M^{T}\right)$

$=\mathrm{I}_{2}-2 \mathrm{MM}^{\mathrm{T}}-2 \mathrm{MM}^{\mathrm{T}}+4 \mathrm{MM}^{\mathrm{T}} \mathrm{MM}^{\mathrm{T}}$

$=\mathrm{I}_{2}-4 \mathrm{MM}^{\mathrm{T}}+4 \mathrm{MM}^{\mathrm{T}}$

$=\mathrm{I}_{2}$

$A X=\lambda X$

$\mathrm{A}^{2} \mathrm{X}=\lambda \mathrm{AX}$

$\mathrm{X}=\lambda(\lambda \mathrm{X})$

$X=\lambda^{2} X$

$\mathrm{X}\left(\lambda^{2}-1\right)=0$

$\lambda^{2}=1$

$\lambda= \pm 1$

Sum of square of all possible values $=2$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 1 February, Shift 2
Subject: Mathematics
Chapter: Matrices
Topic: Characteristic Equation & roots,application of cayley - hamilton theorem