# problem on Trace of the sum of two matrices

• Oct 17th 2012, 07:37 AM
MAX09
problem on Trace of the sum of two matrices
The following question is in the exercise of my text book.

A is 4x4 matrix with 1,-1,2 and -2 as its four eigen values. If B = A^4 - 5A^2 + 5I, which of the following is true?
a) det(A+B)= 0 b) trace (A-B) = 0 c) det(B) = 1 d) trace of (A+B) = 4.

I know for sure that trace of A is 0 as the sum of the eigen values is the same.
But i'm not sure how to arrive at the conclusion for A-B without a little more help in defining the matrix B.
• Oct 17th 2012, 08:10 AM
FernandoRevilla
Re: problem on Trace of the sum of two matrices
Quote:

Originally Posted by MAX09
A is 4x4 matrix with 1,-1,2 and -2 as its four eigen values. If B = A^4 - 5A^2 + 5I, which of the following is true? a) det(A+B)= 0 b) trace (A-B) = 0 c) det(B) = 1 d) trace of (A+B) = 4.

$\displaystyle A$ is diagonalizable and there exists $\displaystyle P$ invertible such that $\displaystyle A=PDP^{-1}$ with $\displaystyle D=\mbox{diag }(1,-1,2,-2)$. Then,

$\displaystyle A+B=P(D+D^4-5D^2+5I)P^{-1}=P\;\mbox{diag }(2,0,3,-1)\;P^{-1}$

Using that similar matrices have the same trace, you'll obtain $\displaystyle \mbox{trace}(A+B)=4$.
• Oct 17th 2012, 08:29 AM
topsquark
Re: problem on Trace of the sum of two matrices
Quote:

Originally Posted by FernandoRevilla
$\displaystyle A$ is diagonalizable and there exists $\displaystyle P$ invertible such that $\displaystyle A=PDP^{-1}=D$

I'm a bit confused here. You have a matrix P such that A = PDP^{-1} and D = PDP^{-1}. Which means A = D and P is the identity.

Is this a typo or am I missing something?

-Dan
• Oct 17th 2012, 08:58 AM
FernandoRevilla
Re: problem on Trace of the sum of two matrices
Quote:

Originally Posted by topsquark
Is this a typo or am I missing something?

A typo, thanks. I've just changed it.
• Oct 17th 2012, 09:40 AM
HallsofIvy
Re: problem on Trace of the sum of two matrices
Since the only information you are given is the eigenvalues of the matrix, any matrix having those eigenvalues must give the same answer. So I would take, say, $\displaystyle A= \begin{bmatrix}1& 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & -2\end{bmatrix}$.

Then it is very easy to find B!
• Oct 17th 2012, 09:43 AM
HallsofIvy
Re: problem on Trace of the sum of two matrices
Since the only information you are given is the eigenvalues of the matrix, any matrix having those eigenvalues must give the same answer. So I would take, say, $\displaystyle A= \begin{bmatrix}1& 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & -2\end{bmatrix}$.

Then it is very easy to find B. (The fact that the characteristic equation of A is $\displaystyle (1- \lambda)(-1+ \lambda)(2- \lambda)(-2- \lambda)= \lambda^4- 5\lambda^2+ 4$ guarentees that!)
• Oct 17th 2012, 10:35 AM
MAX09
Re: problem on Trace of the sum of two matrices
@hallsofIvy

I actually tried the same method, but for some stupid reason, I made a calculation error and ended up getting the wrong answer for A^4 i got a diagonal matrix with {1, 1, 64, 64} which was the most stupid mistakes one could ever make. I retried the same procedure after i got your suggestion and VOILA!! there it is!!

Thanks HallsofIvy
• Oct 17th 2012, 10:36 AM
MAX09
Re: problem on Trace of the sum of two matrices
@FernandoRevilla

Your solution was as elegant as it can ever get!!! Thanks for opening my mind to new ideas!!! I'm working on the solution you had suggested! THanks... :)