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")` and `I` is the unit matrix of order 3.
`|(a_(11)+4,a_(12),a_(13)),(a_(21),a_(22)+4,a_(23)),(a_(31),a_(32),a_(33)+4)|+5 lambda|(a_(11)+1,a_(12),a_(13)),(a_(21),a_(22)+1,a_(23)),(a_(31),a_(32),a_(33)+1)|=0`
then the value of `lambda` 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`