GATE (TF) Textile 2020 Question Paper Solution | GATE/2020/TF/01

For the matrix $\begin{bmatrix} 1 & 1 & 2\\ 0 & 1 & 3\\ 0 & 0 & 1 \end{bmatrix}$ , the eigenvalues of the matrix $A^{2}$ are

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

Firstly we will calculate eigen values of the matrix A

$A-\lambda \times I=0$

I=Identity matrix

$\begin{bmatrix} 1 & 1 & 2\\ 0 & 1 & 3\\ 0 & 0 & 1 \end{bmatrix}-\lambda \times \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}=0$

$\begin{bmatrix} 1-\lambda & 0 & 0\\ 0 & 1-\lambda & 0\\ 0 & 0 & 1-\lambda \end{bmatrix}$

$(1-\lambda)\times(1-\lambda)^2=0$

$\lambda=1,1,1$ ,These are the eigen values of the matrix A

Then eigen values of the matrix $A^2=\lambda^2, \lambda^2 ,\lambda^2$

Eigen values of the matrix $A^2=1^2 ,1^2 ,1^2$

Eigen values of the matrix $A^2$ =1,1,1 (Ans)