If A is any `mxxn` matrix such that AB and BA are both defined, then B is a matrix of type

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the dimensions of matrix A**:
Given that matrix A is of size \( m \times n \).

2. **Define the dimensions of matrix B**:
Let matrix B be of size \( r \times s \).

3. **Condition for AB to be defined**:
For the product \( AB \) to be defined, the number of columns in A must equal the number of rows in B. Therefore, we have:
\[ n = r \]
This means that the number of rows in B (r) must be equal to the number of columns in A (n).

4. **Condition for BA to be defined**:
For the product \( BA \) to be defined, the number of columns in B must equal the number of rows in A. Therefore, we have:
\[ m = s \]
This means that the number of columns in B (s) must be equal to the number of rows in A (m).

5. **Conclusion about the dimensions of B**:
From the above conditions, we can conclude that:
\[ B \text{ is of size } n \times m \]

### Final Answer:
Thus, the type of matrix B is \( n \times m \).

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