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What are the eigenvalues of the following $2\times 2$ matrix? $$\left( \begin{array}{cc} 2 & -1\\ -4 & 5\end{array}\right)$$

1. $-1$ and $1$
2. $1$ and $6$
3. $2$ and $5$
4. $4$ and $-1$
edited | 545 views

Let the eigen values be $a,b$

Sum of Eigen Values = Trace(Diagonal Sum)

$\implies a+b = 2+5 = 7$

Product of Eigen Values = Det(A)

$\implies a\times b = 6$

Solving these we get eigenvalues as 1 and 6.

Option(B) is Correct.

edited by
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how did you derived this?
+1
these are famous properties..
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will this work as check for matrices of size grater than 2*2
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always, for n*n matrices.
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Why the answers cant be 2 &  5 ?

Because after solving characteristics equation we can also get 2 & 5 apart from 1 & 6.

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@shamim_ahmed how you are getting 2 & 5,

characteristic equation will be

let lambda = x

x -7x+6=0  if you solve this equation you will get (1,6)
or follow the above method ..!

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@ankit my bad. It was a silly calculation mistake. :)
(2-x)(5-x)-4=0 x=1,6