NIMCET 2013 Question Paper with solutions

Question : 1 of 120

Marks: +1, -0

If A =

[\table 1,0,0;0,1,1;0,-2,4]

and

6A^{-1}

A^2

+ cA = dI where

A^{-1}

is A - inverse, I is the identity matrix, then (c, d) is:

Solution:

A =

[\table 1,0,0;0,1,1;0,-2,4]

and

6A^{-1}=A^2+cA=dl

...(i)
Now, multiplying Eq. (i) by A, we get

6l=A^3+cA^2+dA

⇒A^3+cA^2+dA-6l=0

...(ii)
Applying Cayley Hamilton rule

|\table1-λ,0,0;0,1-λ,1;0,-2,4-λ|=0

⇒(1-λ)\{(1-λ)(4-λ)+2\}=0

⇒λ^{3}-6λ^{2}+11λ-6=0

......(iii
On comparing Eqs. (ii) and (iii), we get

c=-6

and d =11

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