If A is a matrix of order `2 xx 3`and B is the matrix of order `3xx5` then what is the order of matrix `(AB)^T` ?

Options

A. `2xx5`
B. `5xx2`
C. `3xx3`
D. `5xx3`

Answer

To find the order of the matrix \((AB)^T\), we will follow these steps:

### Step 1: Determine the order of the product \(AB\)
Given:
- Matrix \(A\) is of order \(2 \times 3\) (2 rows and 3 columns).
- Matrix \(B\) is of order \(3 \times 5\) (3 rows and 5 columns).

The product of two matrices \(A\) (of order \(m \times n\)) and \(B\) (of order \(n \times p\)) results in a new matrix of order \(m \times p\).

Here, since \(A\) has 3 columns and \(B\) has 3 rows, we can multiply them:
- The resulting matrix \(AB\) will have:
- Rows from \(A\) = 2
- Columns from \(B\) = 5

Thus, the order of the matrix \(AB\) is \(2 \times 5\).

### Step 2: Find the order of the transpose \((AB)^T\)
The order of the transpose of a matrix is obtained by swapping its rows and columns.

For the matrix \(AB\) which is of order \(2 \times 5\):
- The transpose \((AB)^T\) will have:
- Rows = 5 (originally columns of \(AB\))
- Columns = 2 (originally rows of \(AB\))

Thus, the order of the matrix \((AB)^T\) is \(5 \times 2\).

### Final Answer
The order of the matrix \((AB)^T\) is \(5 \times 2\).

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