Question-12

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 \(A^{-1} = B\)

• Assertion is true and Reason is true. Reason is the correct explanation for Assertion.
• Assertion is true and Reason is true. Reason is not the correct explanation for Assertion.
• Both Assertion and Reason are false.
• Assertion is true. Reason is False.
• Assertion is false. Reason is true.

Answer

• Assertion is true and Reason is true. Reason is the correct explanation for Assertion.
• Assertion is true and Reason is true. Reason is not the correct explanation for Assertion.
• Both Assertion and Reason are false.
• Assertion is true. Reason is False.
• Assertion is false. Reason is true.

Solution

\(A^2 + A = I \implies A(A + I) = I\). 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.