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Part BCSIR NET December 2023rank-one-eigenvalue

Rank one eigenvalue

Let $A = (a_{ij})$ be the n × n real matrix with $a_{ij} =$ ij for all 1 ≤ i, j ≤ n. If n ≥ 3, which one of the following is an eigenvalue of A?

A.1
B.n
C.n(n+1)/2
D.n(n+1)(2n+1)/6✓

Solution

A = vvᵀ with v = (1, 2, …, n) has rank 1; its only non-zero eigenvalue is $v^{T} v = \sum i^{2} = n (n + 1) (2 n + 1) /6$ .

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: Same characteristic polynomial ⇒ similar

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From Eigenvalues and Canonical Forms › Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

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