PEA 2024 PEA Q9 | ISI MSQE

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2024 PEA Q9

Let $x$ and $y$ be two column vectors of length $3$ such that $\sum_{i}x_{i}y_{i}=1$ . What is the rank of $xy^{T}$ ?

Reveal answer and solution

Answer

Solution

1
The condition $\sum x_iy_i=y^{T}x=1\ne 0$ ensures $x\ne 0$ and $y\ne 0$ . The matrix $xy^{T}$ is a nonzero outer product, and every column is a scalar multiple of $x$ . Hence the column space is one-dimensional and
2
$\operatorname{rank}(xy^{T})=1.$

Answer structure / marking notes

Outer products $uv^T$ have rank $1$ whenever both vectors are nonzero.

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Content note

Imported from public/resources/isi/msqe/solutions/pea/2024/ISI_MSQE_PEA_2024_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.