edited by
5,952 views
21 votes
21 votes

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 by

3 Answers

Best answer
35 votes
35 votes

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
2 votes
2 votes
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:

Related questions

19 votes
19 votes
4 answers
1
gatecse asked Sep 21, 2014
7,183 views
Consider the following system of linear equations : $$2x_1 - x_2 + 3x_3 = 1$$ $$3x_1 + 2x_2 + 5x_3 = 2$$ $$-x_1+4x_2+x_3 = 3$$ The system of equations hasno solutiona uni...
23 votes
23 votes
3 answers
3
go_editor asked Feb 12, 2015
7,473 views
The larger of the two eigenvalues of the matrix $\begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}$ is _______.
25 votes
25 votes
5 answers
4
go_editor asked Sep 29, 2014
5,059 views
Consider the matrix as given below.$$\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 7 \\ 0 & 0 & 3\end{bmatrix}$$Which one of the following options provides the CORRECT values of...