Let `A=([a_(i j)])_(3xx3)` be a matrix such that `AA^T=4Ia n da_(i j)+2c_(i j)=0,w h e r ec_(i j)` is the cofactor of `a_(i j)a n dI` is the unit matrix of order 3. `|a_(11)+4a_(12)a_(13)a_(21)a_(22)+4a_(23)a_(31)a_(32)a_(33)+4|+5lambda|a_(11)+1a_(12)a_(13)a_(21)a_(22)+1a_(23)a_(31)a_(32)a_(33)+1|=0` then the value of `10lambda` is _______.

Answer

0.4

Explanation

Given that `A A^(T)=4I`
`implies |A|^(2)=4`
or `|A|= pm 2`
So `A^(T)=4A^(-1)=4 ("adj A")/(|A|)`
`implies [(a_(11),a_(21),a_(31)),(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_("ij")=4/(|A|) c_("ij")`
`implies -2c_("ij")=4/(|A|) c_("ij")" "("as "a_("ij")+2c_("ij")=0)`
`implies |A|=-2`
Now `|A+4I|=|A+A A^(T)|`
`=|A||I+A^(T)|`
`=-2|(I+A)^(T)|`
`=-2|I+A|`
`implies |A+4I|+2|A+I|=0`,
so on comparing, we get `5 lambda=2 implies lambda=2/5`