The GATE Grind

GATE 2020 IN – Question 13

Engineering Mathematics · Linear algebra and matrices: Matrix algebra, systems of linear equations, consistency and rank, eigenvalues and eigenvectors, tensors · 1 mark · Multiple choice

In $Ax=b$, A is a real $2\times4$ matrix and $b\ne0$. Can multiplying by $A^T$ and inverting $A^TA$ always give a unique x?

  1. Yes, for every such A.
  2. No, a unique solution cannot be obtained this way for any such A.
  3. Yes, provided $A^TA$ is well conditioned.
  4. Yes, provided A is well conditioned.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) No, a unique solution cannot be obtained this way for any such A.

Explanation

The rank of A is at most 2, while $A^TA$ is $4\times4$ and has rank at most 2. It is therefore singular for every such A and cannot be inverted. Moreover, any consistent original system has a null space of dimension at least 2 and thus multiple solutions. A least-norm convention would add an extra criterion, not make the original system unique. Select B.

More Linear algebra and matrices: Matrix algebra, systems of linear equations, consistency and rank, eigenvalues and eigenvectors, tensors questions

All Linear algebra and matrices: Matrix algebra, systems of linear equations, consistency and rank, eigenvalues and eigenvectors, tensors questions →