Let A = I _ 2 - 2M M ^ T , where M is real matrix of order 2 × 1...

Given: A = {I}_{2} - 2M{M}^{T}$</p><p>$⇒ {A}^{2} = \left({I}_{2} - 2M{M}^{T}\right) \left({I}_{2} - 2M{M}^{T}\right)

$⇒ {A}^{2} = {I}_{2} - 2M{M}^{T} - 2M{M}^{T} + 4M{M}^{T}M{M}^{T}$

$⇒ {A}^{2} = {I}_{2} - 4M{M}^{T} + 4M{M}^{T} \left\{\text{as}{M}^{T}M = {I}_{1}\right\}$

$⇒ {A}^{2} = {I}_{2}$

Also, $AX = \lambdaX$

$⇒ {A}^{2}X = \lambdaAX$

$⇒ X = \lambda\left(\lambdaX\right) \left\{\text{as}{A}^{2} = {I}_{2} \text{\text{ and } } AX = \lambdaX\right\}$

$⇒ X = {\lambda}^{2}X$

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

$⇒ {\lambda}^{2} = 1$

$⇒ \lambda = \pm 1$

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