If A is any `m xx n` matrix and B is a matrix such that AB and BA are both defined, then B is a matrix of order

Options

A. `nxxn`
B. `mxxm`
C. `mxxn`
D. `nxxm`

Answer

To find the order of matrix B given that A is an m x n matrix and both products AB and BA are defined, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Order of Matrix A**:
Matrix A is given to be of order m x n, which means it has m rows and n columns.

2. **Understand the Multiplication Condition**:
For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix.

3. **Set Up the Order of Matrix B**:
Let the order of matrix B be p x q (where p is the number of rows and q is the number of columns).

4. **Analyze the Product AB**:
For the product AB to be defined:
- A is m x n.
- B must have n rows (to match A's columns).
- Therefore, the order of B must be n x q, where q can be any number of columns.

5. **Analyze the Product BA**:
For the product BA to be defined:
- B is p x q.
- A is m x n.
- For BA to be defined, the number of columns in B (which is q) must equal the number of rows in A (which is m).
- Thus, we have q = m.

6. **Combine the Results**:
From the analysis of AB, we found that B has to be n x q. From the analysis of BA, we found that q must equal m. Therefore, we can conclude that the order of matrix B is n x m.

### Final Answer:
The order of matrix B is n x m.