Problem-10

inverse

Question

Assertion: If \(A^2 + A = I\), then \(A^{-1} = A + I\)

Reason: If \(A\) is an \(m \times n\) matrix, \(B\) is an \(n \times p\) matrix, with \(AB = I\), then \(A^{-1} = B\)

  1. Assertion is true and Reason is true. Reason is the correct explanation for Assertion.

  2. Assertion is true and Reason is true. Reason is not the correct explanation for Assertion.

  3. Both Assertion and Reason are false.

  4. Assertion is true. Reason is False.

  5. Assertion is false. Reason is true.

Solution

Option-(d) is the correct answer.

\(A^2 + A = I \implies A(A + I) = I\). Since \(A\) is a square matrix, \(AB = I \implies B = A^{-1}\). So the Assertion is true. The Reason is however a false statement. It is true only if \(A\) and \(B\) are square matrices. In its current form \(A\) and \(B\) are not necessarily square.