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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 | 790 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.

by Boss (15.5k points)
edited by
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how did you derived this?
+2
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 ..!

+1
@ankit my bad. It was a silly calculation mistake. :)
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Important properties of Eigen values:-

$(1)$Sum of all eigen values$=$Sum of leading diagonal(principle diagonal) elements=Trace of the matrix.

$(2)$ Product of all Eigen values$=Det(A)=|A|$

$(3)$ Any square diagonal(lower triangular or upper triangular) matrix eigen values are leading diagonal (principle diagonal)elements itself.

Example$:$$A=\begin{bmatrix} 1& 0& 0\\ 0&1 &0 \\ 0& 0& 1\end{bmatrix}$

  Diagonal matrix

Eigenvalues are $1,1,1$

$B=\begin{bmatrix} 1& 9& 6\\ 0&1 &12 \\ 0& 0& 1\end{bmatrix}$

Upper triangular matrix

Eigenvalues are $1,1,1$

$C=\begin{bmatrix} 1& 0& 0\\ 8&1 &0 \\ 2& 3& 1\end{bmatrix}$

Lower triangular matrix

Eigenvalues are $1,1,1$

(2-x)(5-x)-4=0 x=1,6
by Boss (14.3k points)

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