# Solving for Eigenvalues

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A certain 4x4 real matrix is known to have these properties:

1) Two of the eigenvalues of A are L1= 3 and L2= 2.

2) The number 3 is an eigenvalue of the matrix A + 2I.

3) det. A = 12.

What are the other 2 eigenvalues of A and what is the characteristic polynomial of A, At, and A-1?

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Eigenvalues are solved. The solution is detailed and well presented.

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A certain 4x4 real matrix is known to have these properties:

1) Two of the eigenvalues of A are L1= 3 and L2= 2.

2) The number 3 is an eigenvalue of the matrix A + 2I.

3) det. A = 12.

What are the other 2 eigenvalues of A and what is the characteristic polynomial of A, At, and A-1?

Solution. Assume that is the characteristic polynomial of A. By hypothesis, we know that 3 ...

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- BSc , Wuhan Univ. China
- MA, Shandong Univ.

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- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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