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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. 21D. `-23` |
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Answer» Correct Answer - 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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