# GATE Papers >> Mechanical >> 2017 >> Question No 128

Question No. 128

Consider the matrix A=$\left[\begin{array}{cc}50& 70\\ 70& 80\end{array}\right]$ whose eigenvectors corresponding to eigenvalues ${\lambda }_{1}$ and ${\lambda }_{2}$ are ${\mathrm{X}}_{1}=\left[\begin{array}{cc}70& \\ {\lambda }_{1}& -50\end{array}\right]$ and ${\mathrm{X}}_{2}=\left[\begin{array}{cc}{\lambda }_{2}& -80\\ 70& \phantom{\rule{0ex}{0ex}}\end{array}\right]$ , respectively. The value of x1T x2 is___________

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Solution of Question No 128 of GATE 2017 Mechanical Paper

$\mathrm A=\begin{bmatrix}50&70\\70&80\end{bmatrix}$

For eigen values

Characteristic equation $\left|\mathrm A-\mathrm{λI}\right|=0$

$\begin{array}{l}\begin{vmatrix}50-\mathrm\lambda&70\\70&80-\mathrm\lambda\end{vmatrix}=0\\\mathrm\lambda^2-130\mathrm\lambda-900=0\\\mathrm{Let}\;{\mathrm\lambda}_1\;\mathrm{and}\;{\mathrm\lambda}_2\;\mathrm{are}\;\mathrm{two}\;\mathrm{Roots}={\mathrm\lambda}_1+{\mathrm\lambda}_2=130\\\mathrm{Now},\\\mathrm x_1^\mathrm T\cdot{\mathrm x}_2=\left[70\;\;{\mathrm\lambda}_1\;-50\right]\begin{bmatrix}{\mathrm\lambda}_2\;-80\\70\end{bmatrix}\\\mathrm x_1^\mathrm T\cdot{\mathrm x}_2=70\left({\mathrm\lambda}_1\;+{\mathrm\lambda}_2\right)-9100=0\;\;\;\mathrm{since}\;{\mathrm\lambda}_1+{\mathrm\lambda}_2=130\end{array}$