The Gateway to Computer Science Excellence
First time here? Checkout the FAQ!
+37 votes

Let A be a $4 \times 4$ matrix with eigen values -5,-2,1,4. Which of the following is an eigen value of  the matrix$\begin{bmatrix} A & I \\ I & A \end{bmatrix}$, where $I$ is the $4 \times 4$ identity matrix?

  1. $-5$
  2. $-7$
  3. $2$
  4. $1$
asked in Linear Algebra by (47 points)
edited by | 3k views
some one explain it in better way .

By Cayley Hamilton theorem:- Every square matrix satisfy it's own characteristic equation.

3 Answers

+34 votes
Best answer

Ans is (C) 2

$Ax = λx$, where $λ$ is the eigen value of $A$ . Hence $(A−λI)x = 0$ or $|A−λI| = 0$
So, for our given matrix, we have

$\begin{bmatrix} A-\lambda I &I \\ I& A - \lambda I \end{bmatrix} = 0$

This is a $ 2 \times 2$ block matrix where the first and last and the second and third elements are the same. So, applying the formula for determinant of a block matrix as given here

When A = D and B = C, the blocks are square matrices of the same order and the following formula holds (even if A and B do not commute)

$det \begin{pmatrix} A & B\\ C& D \end{pmatrix} = det (A-B) \ det(A+B)$

(second last case)

we get,

$|A−\lambda I− I| \times |A-\lambda I + I|= 0 $

$\implies |A-(\lambda+1)I| \times |A - (\lambda -1)I| = 0$

Each of the eigen value of $A$ is the solution of the equation $|A- \alpha I |= 0$ ($\alpha$ being the eigen value of $A$). So, we can equate $\lambda +1$ and $\lambda -1$ to any of the eigen value of $A$, and that will get our value of $\lambda$. If we take $\alpha = 1$, we get $\lambda = 2$, and that is one of the choice. For no other choice, this equation holds. So, (c) 2 is the answer.

answered by Loyal (5.8k points)
edited by

From here, how we get λ=−1   or  λ=1?
Yes, that's a mistake. In hurry I did it. Thanks.
I have edited the answer, i think this makes sense to me, is there anything i am missing?
Is there any easy way to solve this question ? can anyone tell me please ?
Please update the second line.-->"λ is the eigen value of A", rather it should be -"

λ is the eigen value of the matrix A I | I  A
+51 votes

We find the eigenvalues of given matrix by solving its characteristic equation :

|(A – x I)2 – I2| = 0
|(A – (x – 1) I) * (A – (x + 1) I)| = 0
|(A – (x – 1) I)| * |(A – (x + 1) I)| = 0
So, |(A – (x – 1) I)| = 0 or |(A – (x + 1) I)| = 0

Let y be eigenvalues of A, then |(A – y I)| = 0 .

So, by comparing equations we get,
either x – 1 = y or x + 1 = y

Therefore , x = y + 1 or x = y – 1

y = -5, -4, 1, 4 (given)

So, x = −4, −1, 2, 5, −6, −3, 0, 3

Thus, option (C) is the answer.

answered by Active (4.7k points)
Is this method of expantion or soln of det is aplicable for all types of Block matrices or only for this?
Which of the following is an eigen value of  the matrix  [A  I,       where I is the 4×4 identity matrix?

                                                                                       I  A  ]

what does that mean ? inside this 4 * 4 matrix there is I which is again a 4 *4 matrix ?

might be very silly

but this line is confusing me

NO , not for every matrix

Can be used only for matrices  where the first and last and the second and third elements are the same
A is 4x4.

I is 4X4.

And  the representation that is given would be of 8 x8.

why then a 2*2 matrix is taken and solved


in the selected solution the second method is used

in this solution, first method is used 

see the text from wiki

the second methd is to be used
+2 votes
Characteristics equation of given matrix is:

$(A- \lambda)^2 - I^2 = 0$

$(A-\lambda + I)(A- \lambda -I)=0$

$\lambda = A+I, A-I$

Given eigen values of $A=-5,-2,1,4$

Then $A+I$ and $A-I$ eigen values of given matrix are: (according to property of Cayley Hamilton theorem):


answered by Boss (14.6k points)

Related questions

+1 vote
0 answers
asked Apr 28 in Linear Algebra by Hirak Active (3.3k points) | 39 views
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
49,807 questions
54,504 answers
74,887 users