Which of the following are correct?
(A) For a square matrix A, if \(A^3 = I\), then \(A^{-1} = A^2\).
(B) The determinant of only a square matrix can be defined.
(C) If A is a square matrix of order 3, then the number of minors of the matrix A is 3.
(D) If A and B are two non-singular matrices of the same order, then \((AB)^{-1} = B^{-1}A^{-1}\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
(A) For a square matrix A, if \(A^3 = I\), then \(A^{-1} = A^2\). — Correct. \(A^3 = I\) can be written as
\( A \cdot A^2 = A^2 \cdot A = I \)
so by the definition of the inverse, \(A^{-1} = A^2\).
(B) The determinant of only a square matrix can be defined. — Correct. A determinant is defined only for a square matrix.
(C) If A is a square matrix of order 3, then the number of minors of the matrix A is 3. — Incorrect. Every element has its own minor, so a 3 × 3 matrix has 9 minors.
(D) If A and B are two non-singular matrices of the same order, then \((AB)^{-1} = B^{-1}A^{-1}\). — Correct. This is the reversal law:
\( (AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = AA^{-1} = I \)