GATE (TF) Textile 2018 Question Paper Solution | GATE/2018/TF/43

Question 43 (Textile Technology & Fibre Science)

If $A = \begin{bmatrix} 3 & 1\\ 1 & 3\end{bmatrix}$ , then the sum of all eigenvalues of the matrix $M=A^2 -4A^{-1}$ is equal to ___17__.

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$A = \begin{bmatrix} 3 & 1\\ 1 & 3\end{bmatrix}$
To calculate eigen value-
$\left |A-\lambda I \right |=0$
$\left |\begin{bmatrix} 3 & 1\\ 1 & 3\end{bmatrix}-\lambda \begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix} \right |=0$
$\left | \begin{bmatrix} 3-\lambda & 1\\ 1 & 3-\lambda\end{bmatrix}\right |=0$
$[(3-\lambda)(3-\lambda)-1]=0$
$[(9-\lambda^2-6\lambda-1)]=0$
$[(-\lambda^2-6\lambda+8)]=0$
$(\lambda^2-4\lambda-2\lambda+8)=0$
$(\lambda-4)(\lambda-2)=0$
$\lambda=2,4$
These are eigen values of matrix A.
Eigen values of matrix-
$M=A^2 -4A^{-1}$
$M_1=2^2-\frac{4}{2}$
$M_1=2$
and
$M_2=4^2-\frac{4}{4}$
$M_2=15$
Sum of eigen values of this matrix
$M_1+M_2=2+15$
$M_1+M_2=17$ (Ans)