Saved Bookmarks
| 1. |
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 n=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 `3 xx3` matrix B is such that `A^(2) +B^(2) = A +B^(2)A`, then det. `(sqrt(2)BA^(-1))` is equal toA. `1`B. `(1)/(4)`C. `(1)/(2)`D. `16` |
|
Answer» Correct Answer - C 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)` |
|