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Let A=[a("ij")](3xx3) be a matrix such t...

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

Text Solution

Verified by Experts

The correct Answer is:
7
`"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`
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