If A is a matrix of order `2xx3` 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: Identify the orders of matrices A and B
- 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).

### Step 2: Determine the order of the product \(AB\)
- The product of two matrices \(A\) and \(B\) can be performed when the number of columns in \(A\) is equal to the number of rows in \(B\).
- Here, \(A\) has 3 columns and \(B\) has 3 rows, so the multiplication is valid.
- The order of the resulting matrix \(AB\) will be determined by the number of rows from \(A\) and the number of columns from \(B\).
- Therefore, the order of \(AB\) is \(2 \times 5\) (2 rows from \(A\) and 5 columns from \(B\)).

### Step 3: Find the order of the transpose \((AB)^T\)
- The transpose of a matrix switches its rows and columns.
- If the order of matrix \(AB\) is \(2 \times 5\), then the order of \((AB)^T\) will be \(5 \times 2\).

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

---