Which of the following statements are correct?
(A) Inverse of a matrix, if it exists, is unique
(B) \((kA)'=-KA'\) (where \(k\) is any real number)
(C) For an invertible matrix \(A\). \((A^{-1})^{-1}= A\)
(D) For an invertible matrix \(A\). \((A')^{-1}= (A^{-1})'\)
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (C) and (D) only
The statements are standard properties of the inverse and the transpose, where \(AA^{-1} = A^{-1}A = I\) and \((AB)' = B'A'\).
Checking the statements
(A) Inverse of a matrix, if it exists, is unique — Correct. If B and C are both inverses of A, then \(B = B(AC) = (BA)C = C\).
(B) \((kA)'=-KA'\) (where \(k\) is any real number) — Incorrect. Transposing does not change the scalar: \((kA)' = kA'\).
(C) For an invertible matrix \(A\). \((A^{-1})^{-1}= A\) — Correct. \(A^{-1}A = AA^{-1} = I\), so A is the inverse of \(A^{-1}\).
(D) For an invertible matrix \(A\). \((A')^{-1}= (A^{-1})'\) — Correct. \(A'(A^{-1})' = (A^{-1}A)' = I' = I\).