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

Question No. 128 Mechanical | GATE 2017

Consider the matrix A=50707080 whose eigenvectors corresponding to eigenvalues λ1 and λ2 are X1=70λ1-50 and X2=λ2-8070 , respectively. The value of x1T x2 is___________


Answer : 0 to 0


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} $

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