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GATE 2017 CE (CE1) – Question 11

Engineering Mathematics · Linear Algebra: Matrix algebra, systems of linear equations, eigenvalues and eigenvectors · 1 mark · Multiple choice

Matrix $P$ is the inverse of matrix $Q$. If $I$ is the identity matrix, which statement is correct?

  1. $PQ=I$ but $QP\ne I$
  2. $QP=I$ but $PQ\ne I$
  3. $PQ=I$ and $QP=I$
  4. $PQ-QP=I$

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Correct answer: (C) $PQ=I$ and $QP=I$

Explanation

For a nonsingular square matrix, the inverse is two-sided: $Q^{-1}Q=QQ^{-1}=I$. Since $P=Q^{-1}$, both $PQ=I$ and $QP=I$ hold. Their difference is zero, not the identity. Hence C.

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