Let $A = [a_{ij}]_{3×2}$ and $B = [b_{ij}]_{3×4}$ be two matrices. Then the order of the matrix $(A^T. B)^T$ is:

4 × 2

The correct answer is Option (2) → 4 × 2

A is a $3\times 2$ matrix. So $A^{T}$ is a $2\times 3$ matrix.

B is a $3\times 4$ matrix.

Product $A^{T}B$ is defined since inner dimensions match: $(2\times 3)(3\times 4)$.

Thus, $A^{T}B$ is a $2\times 4$ matrix.

Now take transpose:

$(A^{T}B)^{T}$ is of order $4\times 2$.

The order of $(A^{T}B)^{T}$ is $4\times 2$.