GATE (TF) Textile 2009 Question Paper Solution | GATE/2009/TF/22

If the determinant and trace of a 2×2 matrix $M$ are -1 and 0 respectively, then the determinant of $M+M^{-1}$ is

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Option A is correct

Given-

Determinant of 2×2 matrix M is -1

As we know ,the determinant of the matrix=product of the eigen values

Let ,eigen values are a and b

So, -1=a x b
ab=-1

And

Trace of a 2×2 matrix M is 0

As we know, trace of the matix=sum of the eigen values

0=a+b

a=-b

i.e.

a x -a=-1

$a^2=1$

a=1 and b=-1

so, eigen values of matrix M are 1 & -1.

and eigen values of matrix $M^-1$ are 1 & -1

Eigen values of matrix $M+M^-1$ are (1+1) and (-1-1) i.e. 2 & -2
Determinant of $M+M^-1=2\times -2$ =-4 (Ans)

Option A is correct.