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GATE 2016 EE – Question 39

Engineering Mathematics · Linear Algebra: Matrix Algebra · 2 marks · Multiple choice

Let $A$ be a $4 \times 3$ real matrix with rank 2. Which one of the following statement is TRUE?

  1. Rank of $A^T A$ is less than 2.
  2. Rank of $A^T A$ is equal to 2.
  3. Rank of $A^T A$ is greater than 2.
  4. Rank of $A^T A$ can be any number between 1 and 3.

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Show answer and explanation

Correct answer: (B) Rank of $A^T A$ is equal to 2.

Explanation

For a real matrix $A$, the products $A^T A$ and $A$ have the same null space, so they have the same rank. Hence $A^T A$ has rank 2.