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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 $-1$
in Linear Algebra
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3 Answers

25 votes
 
Best answer

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
0
how did you derived this?
2
these are famous properties..
0
will this work as check for matrices of size grater than 2*2
1
always, for n*n matrices.
0
Why the answers cant be 2 &  5 ?

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

Thanks in advance.
0

@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. :)
2

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 $1,1,1$

4 votes
(2-x)(5-x)-4=0 x=1,6
1 vote
Let $\lambda$ be the eigen value.

then,

$\begin{vmatrix} 2- \lambda &-1 \\ -4 & 5- \lambda \end{vmatrix}=0$

$\implies \lambda^2 -7 \lambda +10 -4 = 0$

$\implies \lambda^2 -7 \lambda +6 = 0$

If we substitute the options then only Option $C.$ will satisfy.

$\therefore$ Option $C.$ is the correct answer.
Answer:

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