Question

Let \( A \) be a matrix of order \( m \times n \) and \( B \) be a matrix such that \( A^T B \) and \( B A^T \) are defined. Then, the order of \( B \) is:

Hint:

When determining the order of a matrix \( B \) involved in matrix multiplication, ensure the number of rows and columns of \( B \) matches the dimensions required for the given matrix products.

Answer 1

We are given that \( A \) is a matrix of order \( m \times n \), i.e., \( A \) has \( m \) rows and \( n \) columns. The matrix multiplication \( A^T B \) is defined, which means the number of columns of \( A^T \) (which is \( m \)) must match the number of rows of \( B \). Therefore, \( B \) must have \( m \) rows. Next, the matrix multiplication \( B A^T \) is also defined, which means the number of columns of \( B \) (which is \( n \)) must match the number of rows of \( A^T \) (which is \( n \)). Therefore, \( B \) must have \( n \) columns. Thus, the order of matrix \( B \) must be \( n \times m \). Therefore, the correct answer is \( \boxed{n \times m} \).