If A is matrix of order `mxxn` and B is a matrix such that AB' and B'A are both defined , then order of matrix B is

Options

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

Explanation

To determine the order of matrix B given that A is a matrix of order \( m \times n \) and both \( AB' \) and \( B'A \) are defined, we will follow these steps:

### Step 1: Understand the order of matrix A
Matrix A is given to be of order \( m \times n \). This means it has \( m \) rows and \( n \) columns.

### Step 2: Analyze the multiplication \( AB' \)
For the multiplication \( AB' \) to be defined, the number of columns in matrix A must equal the number of rows in matrix \( B' \) (the transpose of B). Since A has \( n \) columns, we can conclude that:
- The number of rows in \( B' \) must be \( n \).

### Step 3: Determine the order of matrix \( B' \)
Since \( B' \) is the transpose of matrix B, if \( B' \) has \( n \) rows, then matrix B must have \( n \) columns. Let’s denote the order of matrix B as \( p \times q \). From our previous conclusion:
- The number of rows in \( B' \) (which is \( p \)) must be \( n \).

### Step 4: Analyze the multiplication \( B'A \)
Now, we also know that \( B'A \) must be defined. For this multiplication to be defined, the number of columns in \( B' \) must equal the number of rows in A. Since A has \( m \) rows, we can conclude that:
- The number of columns in \( B' \) must be \( m \).

### Step 5: Determine the order of matrix B
Since \( B' \) has \( m \) columns, and we already established that \( B' \) has \( n \) rows, we can conclude the order of \( B' \) is \( n \times m \). Consequently, the order of matrix B, which is the transpose of \( B' \), will be:
- The order of B is \( m \times n \).

### Conclusion
Thus, the order of matrix B is \( m \times n \).

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