📘 Question
If
\[
AB = O
\]
then which of the following is correct?
(a) It is not necessary that either \(A = O\) or \(B = O\)
(b) \(A = O\) or \(B = O\)
(c) \(A = O\) and \(B = O\)
(d) All the above statements are wrong
✏️ Concept Explanation
In matrix algebra, the zero product property does NOT hold like it does in real numbers.
That means:
\[
AB = O \;\nRightarrow\; A = O \text{ or } B = O
\]
It is possible for two non-zero matrices to multiply and give a zero matrix.
Example:
\[
A =
\begin{bmatrix}
1 & -1 \\
1 & -1
\end{bmatrix},
\quad
B =
\begin{bmatrix}
1 & 1 \\
1 & 1
\end{bmatrix}
\]
\[
AB = O
\]
Yet neither \(A\) nor \(B\) is a zero matrix.
✅ Final Answer
\[
\boxed{(a)\; \text{It is not necessary that either } A = O \text{ or } B = O}
\]
💡 Key Concept
Matrix multiplication does not follow the zero product rule. So, \(AB = O\) does not imply either matrix is zero.