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Let A = [a(ij)] be 3 xx 3 matrix given b...

Let `A = [a_(ij)]` be `3 xx 3` matrix given by `a_(ij) = {(((i+j)/(2))+(|i-j|)/(2),if i nej,),((i^(j)-(i.j))/(i^(2)+j^(2)),if i =j,):}`
where `a_(ij)` denotes element of `i^(th)` row and `j^(th)` column of matrix `A`.
On the basis of above information answer the following question:
If `A^(2)+ pA + qI_(3) = 32 A^(-1)`, then `(p +q)` is equal to-

A

`-22`

B

`-20`

C

21

D

`-23`

Text Solution

Verified by Experts

The correct Answer is:
D
We have `A= [(0,2,3),(2,0,3),(3,3,1)]`
`implies |A|= 32 implies A^(-1)` will exist
Also matrix A is non-singular
`therefore` The characteristic equation of matrix A, is
`implies A^(3) - A^(2) - 22A = 32I`
` implies 32A ^(-1) = A^(2) - A - 22I`
`therefore p=-1, q = -22` (on comparing)
`implies(p+q) = - 23`
Also, `A^(2) - A = B^(2) - B^(2)` (Given)
`implies |A||A-I|=|B|^(2)|A-I|`
As `|A-I|ne 0`
`implies |B|^(2) = |A|=32`
`therefore |sqrt(2)BA^(-1)| = (2sqrt(2)|B|)/(|A|)`
`(2sqrt(2)(pm4sqrt(2)))/(32)=(pm16)/(32)=pm(1)/(2)`
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