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Try this problem from IIT JAM 2017 exam (Problem 58) and know how to evaluate Eigen value of a Matrix.

## Eigen value of a Matrix | IIT JAM 2017 | Problem 58

Let $\alpha, \beta, \gamma, \delta$ be the eigenvalues of the matrix
$$\begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 2 \end{bmatrix}$$
Then find the value of $\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}$

Linear Algebra

Matrix

Eigen Values

## Check the Answer

But try the problem first…

Answer: $6$

Source

IIT JAM 2017 , Problem 58

## Try with Hints

First hint

Characteristic Equation of a matrix $A$ of order $n$ is defined by $|A-xI|=0$, where $x$ is scalar and $I$ is the identity matrix of order $n$.

The roots of the characteristic equation are called the Eigen Values of that matrix.

Now it is easy. Give it a try.

Second Hint

Let

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

Then its characteristic equation is $|A-xI|=0$ for some scalar $x$ and $I$ the identity matrix of order $4$

i.e., $\begin{vmatrix} -x & 0 & 0 & 0 \\ 1 & -x & 0 & -2 \\ 0 & 1 & -x & 1 \\ 0 & 0 & 1 & 2-x \end{vmatrix} =0$

$\Rightarrow -x\begin{vmatrix} -x & 0 & -2 \\ 1 & -x & 1 \\ 0 & 1 & 2-x \end{vmatrix} =0$

$\Rightarrow -x[-x\{(-x)(2-x)\}-1]-2(1-0)]=0$

$\Rightarrow x^{4}-2 x^{3}-x^{2}+2 x=0$

$\Rightarrow x(x^{3}-2 x^{2}-x+2)=0$

$\Rightarrow x(x-1)(x-2)(x+1)=0$

Final Step

Then the eigen values are : $0,1,-1,2$

Then, $\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2} =0^2+1^2+(-1)^2+2^2=6$ [ANS]