Isn't 100000 such a huge number (Part-10)?
Consider the matrix
If the sum of the eigenvalues can be represented as and the product of the eigenvalues can be represented as .
Find the value of
The answer is 3.
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Sum of Eigen values is the trace of matrix, thus, a + i b = i + 1 0 0 0 0 0 − i = 1 0 0 0 0 0 + 0 i
Product of eigen values is determinant. d e t ( Y ) = ∣ ∣ ∣ ∣ ∣ ∣ i 0 2 i 0 1 0 0 0 0 0 0 − 2 i 0 − i ∣ ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ − 3 i 0 0 0 1 0 0 0 0 0 0 − 2 i 0 − i ∣ ∣ ∣ ∣ ∣ ∣
Now determinant of diagonal matrix is just the product of diagonal elements, thus, d e t ( Y ) = − 3 0 0 0 0 0 + 0 i
Substituting values, 1 0 0 0 0 0 − ( − 3 0 0 0 0 0 ) = 3