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.

Re: problem on Trace of the sum of two matrices

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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 $\displaystyle \mbox{trace}(A+B)=4$.

Re: problem on Trace of the sum of two matrices

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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

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.

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!

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!)

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

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... :)