Let \(A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}\) and let I be the identity matrix of order 2. Which of the following statements are correct?
(A) \(A^2 = -I\)
(B) \(A^3 = A\)
(C) \(A^4 = I\)
(D) \(A^6 = -I\)
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
Multiplying rows of A by columns of A:
\( \displaystyle A^2 = \begin{bmatrix} 1 - 2 & -1 + 1 \\ 2 - 2 & -2 + 1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} = -I \)
Every higher power now follows from \(A^2 = -I\), since \(-I\) behaves like the number −1 in products:
\(A^3 = A^2 A = -A\)
\(A^4 = (A^2)^2 = (-I)^2 = I\)
\(A^6 = (A^2)^3 = (-I)^3 = -I\)
So \(A^2 = -I\), \(A^3 = -A\), \(A^4 = I\) and \(A^6 = -I\).
Checking the statements
(A) \(A^2 = -I\) — Correct. Direct multiplication gives \(-I\).
(B) \(A^3 = A\) — Incorrect. \(A^3 = -A\), and \(A \ne O\).
(C) \(A^4 = I\) — Correct. \((-I)^2 = I\).
(D) \(A^6 = -I\) — Correct. \((-I)^3 = -I\).