GATE CSE
First time here? Checkout the FAQ!
x
+2 votes
135 views

Let A be (3×3) real matrix. Suppose 1 & -1 are two of three Eigen values of A and 18 is one of the Eigen values of A2+3A. Then

 [ A ]     A is invertible as well as  A2  + 3A is invertible
[ B ]

A^2+3A is invertible but A is not invertible

 

[ C ]

A is invertible but A^2+3A is not invertible

 

[ D ]

Both A & A^2+3A are not invertible

 
 
asked in Linear Algebra by Junior (811 points) 3 17 31
edited by | 135 views

1 Answer

+5 votes
Best answer

Here we have to recall the property :

a)  If  a matrix A is modified  A = f(A) then the individual eigen values will also be updated in the same manner 

b) The product of  eigen values will give us the determinant of the matrix.

c) If the determinant of the matrix is non zero then it is said to be invertible matrix.

d) In an n * n matrix , at most n distinct eigen values possible.

In this context a) point  means ,

If say x is the eigen value of A , then the corresponding eigen value

for  A2  +  3A     will   be    :   x2  + 3x

So given that one of the eigen values of A+ 3A  is 18..So using a) property mentioned above , we have :

           x+ 3x = 18

==>     x  =  3

So 3 is the corresponding eigen value of original matrix..And two of the eigen values already given : 1 , -1

So determinant of matrix A  =  3 * 1 * -1

                                         =  -3  which is non zero  

Hence A is invertible..

Now for A2  + A  matrix we need to calculate other 2 eigen values which can be found using the eigen values 1 and -1 of A.

So eigen value of the new matrix corresponding to the eigen value 1  =  12  + 3(1)   = 4

eigen value of the new matrix corresponding to the eigen value  -1   =  (-1)2  + 3(-1)   =  1 - 3 = -2

So product of eigen values of new matrix =  18 * 4 * -2   =  -144

Hence determinant of new matrix   =   -144 which is non zero as well

So the new matrix is also invertible.

Hence A is invertible as well as  A2  + 3A is invertible.

Hence A) is the correct option.

 

answered by Veteran (89.1k points) 15 58 295
selected by


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
Top Users Oct 2017
  1. Arjun

    23642 Points

  2. Bikram

    17188 Points

  3. Habibkhan

    8734 Points

  4. srestha

    6404 Points

  5. Debashish Deka

    5478 Points

  6. jothee

    5098 Points

  7. Sachin Mittal 1

    4882 Points

  8. joshi_nitish

    4478 Points

  9. sushmita

    4008 Points

  10. Rishi yadav

    3960 Points


Recent Badges

Popular Question Bikram
Regular Mr.OOPs
Avid Reader Venkat Sai
Popular Question jothee
Verified Human anchal Singh
Great Question Kathleen
Verified Human Ankita_ Pawar
Avid Reader kenzou
Notable Question Sumit Chaudhary 2
Nice Comment rahul sharma 5
27,380 questions
35,231 answers
84,395 comments
33,389 users