GATE Electrical Engineering (EE) 2019 Solved Paper

Question : 58 of 65

Marks: +1, -0

Consider a

2 × 2

matrix

{M}=[ {v}_{1} {v}_{2}]

, where,

{v}_{1}

and

{v}_{2}

are the column vectors. Suppose

{M}^{-1-}[\table {u}_{1}^{ {T}} ; {u}_{2}^{ {T}}]

, where

{u}_{1}^{ {T}}

and

u_{2}^{ {T}}

are the row vectors, Consider the following statements:
Statement

1: u_{1}^{ {T}} {v}_{1}=1

and

u_{2}^{ {T}} {v}_{2}=1

Statement 2:

u_{1}^{ {T}} v_{2}=0

and

u_{2}^{ {T}} v_{1}=0

Which of the following options is correct?

Solution:

M=(v_{1} v_{2})=(\table a , c ; b , d)

M^{-1}=(\table u_{1}^{T} ; u_{2}^{T})=(\table e , f ; g , h)

{M}^{-1} {M}= {I}

⇒(\table \;\text " ea "\;+\;\text " bf "\; , \;\text " ec "\;+\;\text " fd "\; ; \;\text " ga "\;+\;\text " hb "\; , \;\text " gc "\;+\;\text " hd "\;)=(\table 1 , 0 ; 0 , 1)

{ea}+ {bf}+1 ⇒ {u}_{1}^{ {T}} {v}_{1}=1

{ec}+ {fd}+=0 ⇒ {u}_{1}^{ {T}} {v}_{2}=0

{ga}+ {hb}=0 ⇒ {u}_{2}^{ {T}} {v}_{1}=0

{gc}+ {hd}=1 ⇒ {u}_{2}^{ {T}} {v}_{2}=1

∴

Both the statements are correct.

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