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+7 votes
545 views

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 $-1$
asked in Linear Algebra by Boss (18.2k points)
edited by | 545 views

2 Answers

+16 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.

answered by Boss (15.5k points)
edited by
0
how did you derived this?
+1
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. :)
+3 votes
(2-x)(5-x)-4=0 x=1,6
answered by Boss (14.3k points)


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