Skip to content
GATEiit

GATE DA 2025 Q28: Properties of A = Iₙ + xxᵀ with xᵀx = 1

Question

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

Try it: multiple select: one or more correct options

Select one or more options
Select all correct options. No negative marking, no partial marks.
Show answer and explanation

Answer: A, B (official GATE 2025 final answer key)

Concept used: Eigenvalues of a rank-one update of the identity matrix.

First step: Find the eigenvalues: Ax=x+x(x⊤x)=2xAx = x + x(x^{\top}x) = 2x, and any vv orthogonal to xx gives Av=vAv = v. So the eigenvalues are 2 (once) and 1 (n−1n - 1 times).

Full step-by-step solution

Detailed working, common mistakes and linked mock tests will be available with a free GATEiit account. Log-in is coming soon.

Share this question

WhatsAppDownload image

Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

More Linear Algebra PYQs

PracticePractise Linear Algebra questions →Full paperAll GATE DA 2025 questions →