Let A = I 2 − 2 M M T , where M is real matrix of order 2 × 1 such that the relation M T M = I 1 holds. If λ…

Given: $A=I_2-2M{M}^T$

$\Rightarrow A^2=\left(I_2-2M{M}^T\right)\left(I_2-2M{M}^T\right)$

$\Rightarrow A^2=I_2-2M{M}^T-2M{M}^T+4M{M}^{T}M{M}^T$

$\Rightarrow A^2=I_2-4M{M}^T+4M{M}^T$ $\left\{\text{as} M^{T}M=I_1\right\}$

$\Rightarrow A^2=I_2$

Also, $AX=\lambda X$

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

$\Rightarrow X=\lambda \left(\lambda X\right)$ $\left\{\text{as} A^2=I_2 \& AX=\lambda X\right\}$

$\Rightarrow X={\lambda }^{2}X$

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

$\Rightarrow {\lambda }^2=1$

$\Rightarrow \lambda =\pm 1$

So, sum of square of all possible values $=2$