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GO Classes Test Series 2024 | Mock GATE | Test 11 | Question: 30

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Let $A$ be a matrix defined as $A=u v^T$, where $u$ and $v$ are column vectors of dimension $3 \times 1$. The resulting matrix $A$ will be of dimension $3 \times 3$. What are the maximum number of nonzero eigenvalues possible for the matrix $A?$
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A = uu*

where u is a 3x1 matrix
so, u* is a 1x3 matrix

The key to this question is AB and BA share non-zero eigenvalues

Lets say, B = u*u
So, B will be a 1x1 matrix

Hence, B will have only 1 eigenvalue

Therefore, Out of 3 eigenvalues of A, only 1 (sharable by B) will be non-zero. Other 2 eigenvalues will be 0.

Answer is 1

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