in Linear Algebra edited by
1,218 views
4 votes
4 votes

The Eigen values of $A=\begin{bmatrix} a& 1& 0\\1 &a &1 \\0 &1 &a \end{bmatrix}$ are______

  1. $a,a,a$
  2. $0,a,2a$
  3. $-a,2a,2a$
  4. $a,a+\sqrt{2},a-\sqrt{2}$
in Linear Algebra edited by
by
1.2k views

2 Comments

edited by

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 $1,1,1$

------------------------------------------------------------------

Apply the above properties to the your question then you will get answer $(d).$

Here $\text{Sum of all eigen values = a+a+a=3a}$ and $\text{Product of all eigen values =|A|=$a^{3}$-2a}$

2
2
Thanks.. :)

I made a BAD silly mistake while finding the determinant... my bad.. !

Thanks a ton.. :)
0
0

1 Answer

6 votes
6 votes
Best answer

Plz see the picture below for detailed answer.

1. Using properties to eliminate options:

2. Using general procedure: 

edited by

4 Comments

Thanks. You can apply properties which will save time. I have added it.
0
0

 @SuvasishDutta 

Good explanation, and always believe in method rather than a shortcut. And property just saves time but for better understanding, the actual method is good enough.

0
0

Yes absolutely right @Lakshman Patel RJIT. 

1
1

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true