If A and B are two matrices such that AB if of order n `xx` n, then which one of the following is correct?

Options

A. A and B should be square matrices of same order.
B. Either A or B should be a square matrix.
C. Both A and B should be of same order.
D. Orders of A and B need not be the same.

Answer

To solve the problem, we need to analyze the conditions under which the product of two matrices \( A \) and \( B \) results in a matrix of order \( n \times n \).

### Step-by-Step Solution:

1. **Understanding Matrix Multiplication**:
- The product of two matrices \( A \) (of order \( m \times p \)) and \( B \) (of order \( p \times q \)) results in a matrix of order \( m \times q \).
- For the product \( AB \) to be defined, the number of columns in \( A \) must equal the number of rows in \( B \).

2. **Given Information**:
- We are given that \( AB \) is of order \( n \times n \).
- This means that the resulting matrix \( AB \) has \( n \) rows and \( n \) columns.

3. **Setting Up the Orders of Matrices**:
- Let’s denote the order of matrix \( A \) as \( m \times p \) and the order of matrix \( B \) as \( p \times q \).
- Since \( AB \) is \( n \times n \), we have:
- \( m = n \) (the number of rows of \( A \) must equal \( n \))
- \( q = n \) (the number of columns of \( B \) must equal \( n \))

4. **Determining the Order of Matrix \( B \)**:
- From the multiplication condition, we know that \( p \) (the number of columns of \( A \)) must equal the number of rows of \( B \).
- Thus, we can conclude that \( p \) can be any positive integer, as long as the conditions for multiplication are satisfied.

5. **Analyzing the Options**:
- **Option 1**: Both \( A \) and \( B \) should be square matrices of the same order. (Not necessarily true)
- **Option 2**: Either \( A \) or \( B \) should be a square matrix. (Not necessarily true)
- **Option 3**: Both \( A \) and \( B \) should be of the same order. (Not necessarily true)
- **Option 4**: The orders of \( A \) and \( B \) need not be the same. (This is correct)

6. **Conclusion**:
- The correct answer is **Option 4**: The orders of \( A \) and \( B \) need not be the same.