Correct Answer - Option 3 : 7 × 2

__Concept:__

- To multiply an ‘m × n’ matrix by an ‘n × p’ matrix, then 'n' must be the same, and the result is an m × p matrix.
- If A be a matrix of order ‘m × n’ then the order of transpose matrix is ‘n × m’

__Calculation:__

Given:

Order of A is 3 × 2, the order of B is 3 × 5 and the order of C is 7 × 5

The transpose of the matrix obtained by interchanging the rows and columns of the original matrix.

So, order of A^{T} is 2 × 3 and order of C^{T} is 5 × 7

Now,

A^{T}B = { 2 × 3 } { 3 × 5 } = 2 × 5

⇒ Order of A^{T}B is 2 × 5

Now order of (A^{T}B)C ^{T} = {2 × 5} {5 × 7} = 2 × 7

∴ Order of [(ATB)CT]T is 7 × 2

Hence, option (3) is correct.