Which of the following statements are correct?
(A) The inverse of an invertible square matrix is unique.
(B) The zero matrix of order 2 is invertible.
(C) The identity matrix I is invertible and \(I^{-1} = I\).
(D) If A is an invertible matrix, then \((A^{-1})^{-1} = A\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
A square matrix A is invertible if there is a square matrix B of the same order such that
\(AB = BA = I\)
and then B is called the inverse of A, written \(A^{-1}\).
If B and C were both inverses of A, then
\(B = BI = B(AC) = (BA)C = IC = C\)
For the zero matrix O, every product OB is O, never I.
So the inverse is unique, O has no inverse, \(II = I\), and \(A^{-1}A = AA^{-1} = I\) shows that A is the inverse of \(A^{-1}\).
Checking the statements
(A) The inverse of an invertible square matrix is unique. — Correct. Two inverses B and C give \(B = B(AC) = (BA)C = C\).
(B) The zero matrix of order 2 is invertible. — Incorrect. \(OB = O \ne I\) for every matrix B.
(C) The identity matrix I is invertible and \(I^{-1} = I\). — Correct. \(II = I\).
(D) If A is an invertible matrix, then \((A^{-1})^{-1} = A\). — Correct. \(A^{-1}A = AA^{-1} = I\).