📘 Question
If
\[
A =
\begin{bmatrix}
2 & -1 & 3 \\
-4 & 5 & 1
\end{bmatrix}
\quad (2 \times 3)
\]
\[
B =
\begin{bmatrix}
2 & 3 \\
4 & -2 \\
1 & 5
\end{bmatrix}
\quad (3 \times 2)
\]
Then:
(a) only AB is defined
(b) only BA is defined
(c) AB and BA both are defined
(d) AB and BA both are not defined
✏️ Step-by-Step Solution
Step 1: Check \(AB\)
\[
A(2 \times 3) \cdot B(3 \times 2)
\]
✔ Inner dimensions match → multiplication possible
Result order:
\[
2 \times 2
\]
—
Step 2: Check \(BA\)
\[
B(3 \times 2) \cdot A(2 \times 3)
\]
✔ Inner dimensions match → multiplication possible
Result order:
\[
3 \times 3
\]
—
Step 3: Conclusion
Both \(AB\) and \(BA\) are defined.
—✅ Final Answer
\[
\boxed{(c)\; \text{AB and BA both are defined}}
\]
—
💡 Key Concept
Matrix multiplication is defined when:
\[
(\text{columns of first}) = (\text{rows of second})
\]