Let `A=[a_(ij)]_(3xx3)` be a matrix such that `A.A^(T)=4I` and `a_(ij)+2c_(ij)=0` where `c_(ij)` is the cofactor of `a_(ij) AAi`& `j`, `I` is the unit matrix of order 3 and `A^(T)` is the transpose of the matrix `A`
If `|(a_(11)+4,a_(12), a_(3)),(a_(21),a_(22)+4,a_(24)),(a_(31),a_(32),a_(33)+4)|+5lamda|(a_(11)+1,a_(12),a_(13)),(a_(21),a_(22)+1,a_(23)),(a_(31),a_(32),a_(33)+1)|=0` then `lamda=a/b` where `a` and `b` are coprime positive integers then the value of `a+b`is______

Answer

7

Explanation

`"AA"^(T)=4limplies|A|=+-2`
`A^(T)=4A^(-1)=(4Adj(A))/(|A|)`
`implies[(a_(11), a_(21), a_(31)),(a_(12), a_(22), a_(32)),(a_(12), a_(22), a_(32)), (a_(13),a_(23),a_(33))]=4/(|A|)[(c_(11),c_(21),c_(31)),(c_(12),c_(22),c_(32)),(c_(13),c_(23),c_(33))]`
Now, `a_(ji)=4/(|A|)=c_(ij)=-2c_(ij)implies|A|=-2`
Now, `|A|+4l|=|A+"AA"^(T)|=|A||l+A^(T)|+=-2|(l+A)^(T)|=-2l|l+A|` so, `(|A+4l|)/(|A+l|)=-2=-5lamda`
`lamda=2/5`