Let A = I n + xx, where I n is the n n identity matrix and x Rn, x x = 1. Which of the following options is/are correct?

$A^{-1}$ has a negative eigenvalue

GATE 2025 DA – Question 28

Linear Algebra · Eigenvalues, eigenvectors and determinant · 1 mark · Multiple select

Let $A = I_n + xx^\top$, where $I_n$ is the $n \times n$ identity matrix and $x \in \mathbb{R}^n$, $x^\top x = 1$. Which of the following options is/are correct?

Rank of $A$ is $n$
$A$ is invertible
0 is an eigenvalue of $A$
$A^{-1}$ has a negative eigenvalue

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) Rank of $A$ is $n$; (B) $A$ is invertible

Explanation

$xx^\top$ has the eigenvalue $x^\top x = 1$ in the direction of $x$ and 0 in every direction perpendicular to $x$. So $A = I_n + xx^\top$ has the eigenvalue 2 once and the eigenvalue 1 for the other $n - 1$ directions. All eigenvalues are positive, so $A$ has full rank $n$ and is invertible (A and B are true) and 0 is not an eigenvalue (C is false). The eigenvalues of $A^{-1}$ are $\frac{1}{2}$ and 1, which are positive (D is false).