GATE Papers >> Mechanical >> 2016 >> Question No 236

Question No. 236 Mechanical | GATE 2016

The number of linearly independent eigenvectors of matrix A=210020003 is _________


Answer : 2 : 2


Solution of Question No 236 of GATE 2016 Mechanical Paper

$ \mathrm A=\begin{bmatrix}2&1&0\\0&2&0\\0&0&3\end{bmatrix} $

Matrix A has Eigen values

$ \begin{array}{l}\mathrm\lambda=2,2,3\\\because\mathrm{Ax}=\mathrm{λx}\end{array} $

Eigen vector of $ \mathrm\lambda=2 $

$ \begin{array}{l}\begin{bmatrix}2&1&0\\0&2&0\\0&0&3\end{bmatrix}\begin{bmatrix}{\mathrm x}_1\\{\mathrm x}_2\\{\mathrm x}_3\end{bmatrix}=2\begin{bmatrix}{\mathrm x}_1\\{\mathrm x}_2\\{\mathrm x}_3\end{bmatrix}\\2{\mathrm x}_1+{\mathrm x}_2=2{\mathrm x}_1=>{\mathrm x}_2=0\end{array} $

And $ 2{\mathrm x}_2=2{\mathrm x}_2 $

And $ 3{\mathrm x}_3=2{\mathrm x}_3\;=>{\mathrm x}_3=0 $

Assume $ {\mathrm x}_1=\mathrm\alpha $

Hence Eigen vector $ =\begin{bmatrix}{\mathrm x}_1\\{\mathrm x}_2\\{\mathrm x}_3\end{bmatrix}=\begin{bmatrix}\mathrm\alpha\\0\\0\end{bmatrix} $

Eigen vector of $ \mathrm\lambda=2 $

$ \begin{array}{l}\begin{bmatrix}2&1&0\\0&2&0\\0&0&3\end{bmatrix}\begin{bmatrix}{\mathrm x}_1\\{\mathrm x}_2\\{\mathrm x}_3\end{bmatrix}=3\begin{bmatrix}{\mathrm x}_1\\{\mathrm x}_2\\{\mathrm x}_3\end{bmatrix}\\2{\mathrm x}_1+{\mathrm x}_2=3{\mathrm x}_1\\{\mathrm x}_2={\mathrm x}_1\end{array} $

And $ 2{\mathrm x}_2=3{\mathrm x}_2=>{\mathrm x}_2=0\;\&\;{\mathrm x}_1=0 $

And $ 3{\mathrm x}_3=3{\mathrm x}_3 $

Assume $ {\mathrm x}_3=\mathrm\alpha $

Hence Eigen vector $ \begin{bmatrix}{\mathrm x}_1\\{\mathrm x}_2\\{\mathrm x}_3\end{bmatrix}=\begin{bmatrix}0\\0\\\mathrm\alpha\end{bmatrix} $

Hence total number of linear independent Eigen vector =2

Alternate method:

Matrix A has Eigen values

$ \begin{array}{l}\mathrm\lambda=2,2,3\\\mathrm{For}\;\mathrm\lambda=2\end{array} $

Characteristic matrix $ \mathrm X=\begin{bmatrix}0&1&0\\0&0&0\\0&0&1\end{bmatrix} $

$ \begin{array}{l}{\mathrm R}_2\leftrightarrow{\mathrm R}_3\\\mathrm X=\begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmatrix}\end{array} $ 

Hence rank of characteristc matrix $ \left[\mathrm A-\mathrm{λI}\right]=2 $

Number of linearly independent vectors $ =\mathrm n-\mathrm r=3-2=1\;\mathrm{For}\;\mathrm\lambda=3 $

Characteristic matrix $ \mathrm X=\begin{bmatrix}-1&1&0\\0&-1&0\\0&0&0\end{bmatrix} $

Hence rank of characteristic matrix $ \left[\mathrm A-\mathrm{λI}\right]=2 $

Number of linearly independent vectors $ =\mathrm n-\mathrm r=3-2=1 $ Total number of linearly independent vectors $=2$

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