GATE Papers >> Mechanical >> 2017 >> Question No 1

Question No. 1 Mechanical | GATE 2017

The product of eigenvalues of matrix P is

$\mathrm P=\begin{bmatrix}2&0&1\\4&-3&3\\0&2&-1\end{bmatrix}$


Answer : (B) 2


Solution of Question No 1 of GATE 2017 Mechanical Paper

$ \mathrm P\begin{bmatrix}2&0&1\\4&-3&3\\0&2&-1\end{bmatrix} $

Product of eigen values = det(p)

$ \det(\mathrm p)=(3-6)+0+1(8)=-6+8=2 $

Product of eigen values = 2

Alternate method

For finding eigen values $ \mathrm\rho-\mathrm{λI}=0 $

$ \begin{array}{l}\begin{vmatrix}2-\mathrm\lambda&0&1\\4&-3-\mathrm\lambda&3\\0&2&-1-\mathrm\lambda\end{vmatrix}=0\\\mathrm\lambda^3+2\mathrm\lambda^2-11\mathrm\lambda-2=0\end{array} $

Compare with $ \mathrm{ax}^3+\mathrm{bx}^3+\mathrm{cx}+\mathrm d=6 $

Product of roots $ =\left(-1\right)^3\frac{\mathrm d}{\mathrm a} $

Product of eigen values $ =\left(-1\right)^3\frac{\left(-2\right)}1=2 $

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