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June 7th, 2016, 02:40 PM  #1 
Newbie Joined: Jun 2016 From: sweden Posts: 2 Thanks: 1  Question about matrix
Given a diagonalizable 2×2 Matrix A, where the sum of the diagonal elements is 2 and where one of the eigenvalues is 3, find all eigenvalues of A$^{1}$. What is meant by "where the sum of the diagonal elements is 2"?
Last edited by skipjack; June 20th, 2016 at 03:24 AM. 
June 7th, 2016, 02:53 PM  #2 
Math Team Joined: Nov 2014 From: Australia Posts: 672 Thanks: 239 
It most likely means the sum of the elements on the main diagonal is 2. That is, if $$A = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right] $$ then the sum of the diagonal elements would be $a + d$. 
June 7th, 2016, 04:50 PM  #3 
Newbie Joined: Jun 2016 From: sweden Posts: 2 Thanks: 1 
From there then: how do I find the determinant? Or do you have some tips for the task? Last edited by skipjack; June 20th, 2016 at 05:06 AM. 
June 7th, 2016, 11:46 PM  #4 
Math Team Joined: Nov 2014 From: Australia Posts: 672 Thanks: 239 
a) If $A = \left[\begin{array}{cc} a & b\\ c & d \end{array}\right]$ then what is $\det A$? b) What can you say about $\det (3I  A)$? c) Can you find $\det A$ using your answer to b)? 
June 19th, 2016, 08:12 PM  #5 
Newbie Joined: Jun 2016 From: Hong Kong Posts: 19 Thanks: 1 
In this case,you can get $\displaystyle \text{tr}(A)=a+d,\,\det(A)=adbc$ to solve this problem. There are two properties for square matrix. 1. the sum of eigenvalues equals the sum of diagonal elements 23=1 the eigenvalues of A are 3,1 2. the eigenvalues of the inverse of A equals the inverse of the eigenvalues of A the eigenvalues of $\displaystyle A^{1}$ are $\displaystyle \frac{1}{3},1$ Last edited by skipjack; June 20th, 2016 at 05:12 AM. 

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