GATE CE 2022 Set 2 — Question 45
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
Let be a non-zero vector of size . Which of the following statement(s) is/are TRUE?
Correct answer
(A) yy^T is a symmetric matrix.; (B) y^Ty is an eigenvalue of yy^T.
Solution
Let be a non-zero vector (where ).
- Statement (A): . Since the matrix equals its transpose, it is symmetric. TRUE.
- Statement (B): Consider the product . Since is a scalar (let ), we have . This is the definition of an eigenvalue equation , where is the eigenvector. Thus, is an eigenvalue. TRUE.
- Statement (C): The rank of an outer product of two non-zero vectors is always 1. Therefore, rank() = 1. FALSE.
- Statement (D): A matrix is invertible if and only if it is full rank. Since the rank is 1 and the dimension is , the matrix is singular (not invertible). FALSE.
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