GATE Data Science and Artificial Intelligence 2024 | Sample Paper | Question: 25

Question

Two eigenvalues of $3 \times 3$ matrix $\mathbf{X}$ are $(1+i)$ and $2$. The determinant of the text matrix $\mathrm{X}$ is __________.

Answer 1

Sol: Note there is an observation missed here. Please read subsequent comments as well.

Comments

No. You are not missing anything now. Term "Text Matrix" should also be defined in the question as it is not a usual term in the data science and machine learning.

It is not “only” for real matrices, complex matrices can also have conjugate pairs for the eigen values. For example if you replace $3+i$ with $1-i$ in the above example, but it does not for hold for all the complex matrices. It holds when you get the real coefficients in the characteristic polynomial.  

You have answered the question of "when" but not "why". So, if you are curious to know why we get the conjugate pairs then you can see the proof of "Complex Conjugate Root Theorem".

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Understood. I will see the proof. Thank you for the explanation.

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Thank you @Riya_23 mam 

Answer 2

1. Product of Eigenvalues: The determinant of a square matrix is equal to the product of its eigenvalues.
2. Complex Conjugate Eigenvalues: For real matrices with complex conjugate eigenvalues like (1 + i) and (1 - i), these eigenvalues contribute their product to the determinant even though they're not real themselves.

Therefore, the determinant of matrix X will be:

$det(X) = (1 + i)(1 - i) * 2$

Expanding the complex conjugate product:

$det(X) = (1^2 + (-i)^2) * 2$

$det(X) = (1 + 1) * 2$

$det(X) = 2 * 2$

$det(X) = 4$

So, the determinant of the matrix X is 4.