If `A` is a matrix of order `mxxn` and `B` is a matrix such that `A B^T` and `B^T A` are both defined, then the order of matrix `B` is

Options

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

Explanation

To find the order of matrix \( B \) given that \( A \) is a matrix of order \( m \times n \) and both \( A B^T \) and \( B^T A \) are defined, we can follow these steps:

### Step 1: Understand the Order of Matrix \( A \)
Matrix \( A \) has an order of \( m \times n \), which means it has \( m \) rows and \( n \) columns.

### Step 2: Define the Order of Matrix \( B \)
Let’s assume matrix \( B \) has an order of \( p \times q \), where \( p \) is the number of rows and \( q \) is the number of columns.

### Step 3: Determine the Order of \( B^T \)
The transpose of matrix \( B \), denoted as \( B^T \), will have the order \( q \times p \) (the rows and columns are interchanged).

### Step 4: Analyze the Multiplication \( A B^T \)
For the multiplication \( A B^T \) to be defined, the number of columns in \( A \) must equal the number of rows in \( B^T \). Therefore, we have:
\[ n = q \]
This means the number of columns in \( A \) (which is \( n \)) must equal the number of rows in \( B^T \) (which is \( q \)).

### Step 5: Analyze the Multiplication \( B^T A \)
Next, for the multiplication \( B^T A \) to be defined, the number of columns in \( B^T \) must equal the number of rows in \( A \). Therefore, we have:
\[ p = m \]
This means the number of rows in \( B \) (which is \( p \)) must equal the number of rows in \( A \) (which is \( m \)).

### Step 6: Conclude the Order of Matrix \( B \)
From the above steps, we have:
- \( p = m \)
- \( q = n \)

Thus, the order of matrix \( B \) is:
\[ B: m \times n \]

### Final Answer
The order of matrix \( B \) is \( m \times n \).