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The trace of a 2 × 2 matrix is 4 and its determinant is 8. If one of the eigenvalues is 2(1 + i), the other eigenvalue is
  • a)
    2(1 – i)
  • b)
    2(1 + i)
  • c)
    (1 + 2i)
  • d)
    (1 – 2i)
Correct answer is option 'A'. Can you explain this answer?
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The trace of a 2 × 2 matrix is 4 and its determinant is 8. If one...
Given information:
- The trace of a 2x2 matrix is 4.
- The determinant of the matrix is 8.
- One of the eigenvalues is 2(1 + i).

Explanation:
To find the other eigenvalue, we can use the following properties of eigenvalues:

1. The sum of the eigenvalues is equal to the trace of the matrix.
2. The product of the eigenvalues is equal to the determinant of the matrix.

Using these properties, we can solve for the other eigenvalue.

Step 1: Finding the sum of eigenvalues
The trace of the matrix is given as 4, which is equal to the sum of the eigenvalues. Let's denote the other eigenvalue as λ2.

Sum of eigenvalues = λ1 + λ2 = 4

Substituting the value of λ1 (given as 2(1 + i)):

2(1 + i) + λ2 = 4

2 + 2i + λ2 = 4

λ2 = 4 - 2 - 2i

λ2 = 2 - 2i

Step 2: Finding the product of eigenvalues
The determinant of the matrix is given as 8, which is equal to the product of the eigenvalues. Let's denote the other eigenvalue as λ2.

Product of eigenvalues = λ1 * λ2 = 8

Substituting the value of λ1 (given as 2(1 + i)) and λ2 (found in Step 1):

2(1 + i) * (2 - 2i) = 8

4 - 4i + 4i - 4i^2 = 8

4 - 4i + 4i + 4 = 8

8 = 8

The product of the eigenvalues is indeed 8.

Conclusion:
From the calculations, we have found that the other eigenvalue is 2 - 2i, which matches option A: 2(1 - i).
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The trace of a 2 × 2 matrix is 4 and its determinant is 8. If one...
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The trace of a 2 × 2 matrix is 4 and its determinant is 8. If one of the eigenvalues is 2(1 + i), the other eigenvalue isa)2(1 – i)b)2(1 + i)c)(1 + 2i)d)(1 – 2i)Correct answer is option 'A'. Can you explain this answer?
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