If A = $\begin{bmatrix$}[r] a & b \end{bmatrix} \text{ and } \begin{bmatrix}[r] c \ d \end{bmatrix}, then :

1. only matrix AB is possible

2. only matrix BA is possible

3. both matrices AB and BA are possible

4. both matrices AB and BA are possible, AB = BA

Order of matrix A = 1 × 2

Order of matrix B = 2 × 1

Since, no. of columns in A is equal to the no. of rows in B and no. of columns in B is equal to the no. of rows in A.

∴ AB and BA are possible.

$AB = \begin{bmatrix$}[r] a \times c + b \times d \end{bmatrix} \\[1em] = \begin{bmatrix}[r] ac + bd \end{bmatrix}. \\[1em] BA = \begin{bmatrix}[r] c \times a & c \times b \ d \times a & d \times b \end{bmatrix} \\[1em] = \begin{bmatrix}[r] ca & cb \ da & db \end{bmatrix}.

∴ AB ≠ BA.

Hence, Option 3 is the correct option.