Matrices Question 2
Question 2 - 2024 (01 Feb Shift 2)
Let $A=I _2-M M^{T}$, where $M$ is real matrix of order $2 \times 1$ such that the relation $M^{T} M=I _1$ holds. If $\lambda$ is a real number such that the relation $AX=\lambda X$ 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 :
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Answer (2)
Solution
$A=I _2-2 MM^{T}$
$A^{2}=\left(I _2-2 MM^{T}\right)\left(I _2-2 MM^{T}\right)$
$=I _2-2 MM^{T}-2 MM^{T}+4 MM^{T} MM^{T}$
$=I _2-4 MM^{T}+4 MM^{T}$
$=I _2$
$AX=\lambda X$
$A^{2} X=\lambda AX$
$X=\lambda(\lambda X)$
$X=\lambda^{2} X$
$X\left(\lambda^{2}-1\right)=0$
$\lambda^{2}=1$
$\lambda= \pm 1$
Sum of square of all possible values $=2$
