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+7 votes
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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 $-1$
asked in Linear Algebra by Boss (18.1k points)
edited by | 506 views

2 Answers

+14 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 (15.3k 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.
+3 votes
(2-x)(5-x)-4=0 x=1,6
answered by Boss (14.2k points)


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