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What are the eigenvalues of the matrix $P$ given below

$$P= \begin{pmatrix} a &1 &0 \\ 1& a& 1\\ 0&1 &a \end{pmatrix}$$                               

  1. $a, a -√2, a + √2$
  2. $a, a, a$
  3. $0, a, 2a$
  4. $-a, 2a, 2a$

8 Answers

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$\begin{bmatrix} a &1 &0 \\ 1& a & 1\\ 0&1 &a \end{bmatrix}$

a can be any number. Let's keep it 1.

 

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

Trace = $3$ (Sum of eigenvalues)

Determinant = $-1$ (Product of eigenvalues)

 

Look at option A:

  • $1,1-\sqrt2,1+\sqrt2$

Sum = $3$

Product = $-1$

 

This is our answer.

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sum of eigen value = sum of the leading diagonal elements= trace of the matrix,

here sum= a+a+a=3a ----all the four options satisfy this condition.

now from the second property --product of eigen value is equal to value of determinat of the matrix.

in this the valu of determinant is a(a^2-2) which is satisfied by option A.

Hence the option A is correct.
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Basically here we find the determinant of matrix and it is equal to=a^3-2*a .now we know that products of eigen values will give the determinant of matrices and sum of eigens value gives the trace of matrix so here only one option A follows the this criteria.
Answer:

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