GATE 2022 CE (CE2) – Question 45

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 2 marks · Multiple select

Let y be a nonzero 2022×1 vector. Which statements are true?

$yy^T$ is symmetric
$y^Ty$ is an eigenvalue of $yy^T$
$yy^T$ has rank 2022
$yy^T$ is invertible

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $yy^T$ is symmetric; (B) $y^Ty$ is an eigenvalue of $yy^T$

Explanation

Transposing $yy^T$ gives the same matrix, so A is true. Also $(yy^T)y=y(y^Ty)$, making the nonzero vector y an eigenvector with eigenvalue yᵀy: B is true.

The outer product has rank 1, leaving 2021 zero eigenvalues. It is singular, so C and D are false. **Answer: A and B**.