Question

Let A and B be two invertible matrices, then which of the following is true ?

• A. \(A^{-1}\) may not invertible
• B. \((A^{-1})^{-1} = A\)
• C. \((AB)^{-1} = A^{-1} B\)
• D. \((A^{-1})^{-1} = A^{-1}\)

Hint:

The inverse operation behaves similarly to the transpose operation:
- Double operations cancel out: \((A^{-1})^{-1} = A\) and \((A^T)^T = A\).
- Product operations reverse the order: \((AB)^{-1} = B^{-1} A^{-1}\) and \((AB)^T = B^T A^T\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A square matrix \(A\) is called invertible if there exists a matrix \(A^{-1}\) such that:
\[ A A^{-1} = A^{-1} A = I \]
This question tests standard properties of matrix inverses.

Step 2: Detailed Explanation:
Let us analyze each of the given statements:
- Option (A): "All inverses are themselves invertible."
If \(A\) is invertible, then its inverse \(A^{-1}\) is also invertible, and its inverse is \(A\).
Therefore, claiming \(A^{-1}\) may not be invertible is incorrect.
- Option (B): "The inverse of an inverse matrix is the original matrix."
By definition, if we take the inverse of \(A^{-1}\), we are looking for a matrix \(B\) such that \(A^{-1} B = B A^{-1} = I\).
Since \(A^{-1} A = A A^{-1} = I\), this matrix \(B\) is indeed \(A\).
Therefore:
\[ (A^{-1})^{-1} = A \]
This is a fundamental property of matrix algebra. Hence, Option (B) is correct.
- Option (C): The inverse of the product of two invertible matrices is the product of their inverses in reverse order:
\[ (AB)^{-1} = B^{-1} A^{-1} \]
Option (C) states \((AB)^{-1} = A^{-1} B\), which is incorrect.
- Option (D): States \((A^{-1})^{-1} = A^{-1}\), which is incorrect because the double inverse cancels out to give \(A\), not \(A^{-1}\).
Therefore, only Option (B) is correct.

Step 3: Final Answer:
The correct option is (B).