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June 7th, 2016, 02:40 PM   #1
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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"?
Thanks from MMath

Last edited by skipjack; June 20th, 2016 at 03:24 AM.
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June 7th, 2016, 02:53 PM   #2
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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 $a + d$.
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June 7th, 2016, 04:50 PM   #3
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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.
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June 7th, 2016, 11:46 PM   #4
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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)?
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June 19th, 2016, 08:12 PM   #5
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In this case,you can get $\displaystyle \text{tr}(A)=a+d,\,\det(A)=ad-bc$ to solve this problem.
There are two properties for square matrix.

1. the sum of eigenvalues equals the sum of diagonal elements

2-3=-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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