Let A, B and C be square matrices of order 2, I the identity matrix and O the zero matrix of order 2. Which of the following statements are correct?
(A) \(AI = IA = A\)
(B) \(AO = OA = O\)
(C) If \(A^2 = O\), then \(A = O\).
(D) If \(AB = AC\) and \(A \ne O\), then \(B = C\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
Multiplying by I keeps every element unchanged, and multiplying by O makes every element 0. Statements (C) and (D) are tested with simple matrices.
For (C), take
\( \displaystyle A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix},\quad A^2 = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
For (D), take
\( \displaystyle A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix},\quad B = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix},\quad C = O \)
\(AB = O = AC\), but \(B \ne C\)
So I and O behave like 1 and 0 in products, but a non-zero matrix can have \(A^2 = O\), and \(AB = AC\) does not force \(B = C\).
Checking the statements
(A) \(AI = IA = A\) — Correct. I is the identity for multiplication.
(B) \(AO = OA = O\) — Correct. Every product element is a sum of terms with a factor 0.
(C) If \(A^2 = O\), then \(A = O\) — Incorrect. \(\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\) is non-zero with square O.
(D) If \(AB = AC\) and \(A \ne O\), then \(B = C\) — Incorrect. With \(A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\), \(B = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}\) and \(C = O\), \(AB = AC = O\).