GATE (TF) Textile 2014 Question Paper Solution | GATE/2014/TF/26

Question 26 (Textile Engineering & Fibre Science)

Of the two eigen values of the matrix $\begin{bmatrix}1 & 2\\ 4 & 3 \end{bmatrix}$

(A)	One is positive, one is negative
(B)	Both are positive
(C)	Both are negative
(D)	Both form a complex conjugate

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Given matrix-
A= $\begin{bmatrix}1 & 2\\ 4 & 3 \end{bmatrix}$
To calculate eigen values of the matrix-
$\left |A-\lambda I \right |=0$
$\begin{bmatrix}1 & 2\\ 4 & 3 \end{bmatrix}-\lambda\times \begin{bmatrix}1 & 0\\ 0 & 1 \end{bmatrix} =0$
$\begin{bmatrix}1 & 2\\ 4 & 3 \end{bmatrix}-\begin{bmatrix}\lambda & 0\\ 0 & \lambda \end{bmatrix}=0$
$\begin{bmatrix}1-\lambda & 2\\ 4 & 3-\lambda \end{bmatrix}=0$
$\left |\begin{bmatrix}1-\lambda & 2\\ 4 & 3-\lambda \end{bmatrix} \right |=0$
$(1-\lambda)(3-\lambda)-8=0$
$(3-4 \lambda+\lambda^2)-8=0$
$(\lambda^2-4\lambda-5)=0$
$(\lambda^2-5\lambda+\lambda-5)=0$
$(\lambda-5)(\lambda+1)=0$
$\lambda=-1 , 5$
These are the eigen values-
i.e. one is positive and one is negative.
Option A is correct.