GATE (TF) Textile 2016 Question Paper Solution | GATE/2016/TF/26

Question 26 (Textile Engineering & Fibre Science)

The eigen values and the eigen vectors of $\begin{pmatrix} 3 & 4 \\ 4 & -3 \end{pmatrix}$ are

(A)	$\pm 5$ and $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ , $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ respectively
(B)	$\pm 3$ and $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ , $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ respectively
(C)	$\pm 4$ and $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ , $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ respectively
(D)	$\pm 5$ and $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$ , $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$ respectively

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Given-
$A=\begin{pmatrix} 3 & 4 \\ 4 & -3 \end{pmatrix}$
To calculate eigen value and eigen vectors-
$A-\lambda I=0$
$\begin{pmatrix} 3 & 4 \\ 4 & -3 \end{pmatrix}-\lambda \times \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}=0$
$\begin{pmatrix} 3 & 4 \\ 4 & -3 \end{pmatrix}-\begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix}=0$
$\begin{pmatrix} 3-\lambda & 4 \\ 4 & -3-\lambda \end{pmatrix}=0$
$\left |\begin{pmatrix} 3-\lambda & 4 \\ 4 & -3-\lambda \end{pmatrix} \right |=0$
$-(9-\lambda^2)-16=0$
$-9+\lambda^2-16=0$
$\lambda^2-25=0$
$\lambda^2=25$
$\lambda=-5 , 5$ (These are eigen values)
Eigen vector at $\lambda=5$
$\begin{pmatrix} 3-5 & 4 \\ 4 & -3-5 \end{pmatrix}$
$\begin{pmatrix} -2 & 4 \\ 4 & -8 \end{pmatrix}$
$\begin{pmatrix} -2 & 4 \\ 4 & -8 \end{pmatrix} \times \begin{pmatrix} x_1 \\ x_2\end{pmatrix}=0$
-2 x1+4 x2=0
4 x1-8 x2=0
By solving-
x1=2 x2
$\begin{pmatrix} 2x_2 \\ x_2\end{pmatrix}$
Eigen vector-
$\begin{pmatrix} 2 \\ 1\end{pmatrix}$ (Ans)
and at $\lambda=-5$
$\begin{pmatrix} 3+5 & 4 \\ 4 & -3+5 \end{pmatrix}$
$\begin{pmatrix} 8 & 4 \\ 4 & 2 \end{pmatrix} \times \begin{pmatrix} x_1 \\ x_2\end{pmatrix}=0$
8 x1+4 x2=0
4 x1+2 x2=0
By solving-
$x_1=\frac{-1}{2} x_2$

$\begin{pmatrix} x_1 \\ x_2\end{pmatrix}$
$\begin{pmatrix} \frac{-1}{2}x_2 \\ x_2\end{pmatrix}$
Eigen vector-
$\begin{pmatrix} -1 \\ 2\end{pmatrix}$ (Ans)