Question

A and B are two non-square matrices. If \( P = A + B \), \( Q = A^TB \), \( R = AB^T \), then the matrices whose order is equal to the order of A are

• A. PQ and QR
• B. RQ and QP
• C. PQ and RP
• D. PQR and RPQ

Hint:

Write down dimensions explicitly: \( A_{m \times n} \). Matrix multiplication \( X_{a \times b} \cdot Y_{c \times d} \) is valid only if \( b = c \), and the result is \( a \times d \).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:

We need to determine the dimensions (order) of the resulting matrices after multiplication and addition.
Step 2: Detailed Explanation:

Let the order of matrix \( A \) be \( m \times n \). Since \( A \) is non-square, \( m \neq n \). Since \( P = A + B \) is defined, \( A \) and \( B \) must have the same order. So, Order of \( B = m \times n \). Consequently, Order of \( P = m \times n \). Now determine the orders of \( Q \) and \( R \): 1. \( Q = A^T B \) * \( A^T \) is \( n \times m \). * \( B \) is \( m \times n \). * Order of \( Q = (n \times m) \times (m \times n) = n \times n \) (Square matrix). 2. \( R = A B^T \) * \( A \) is \( m \times n \). * \( B^T \) is \( n \times m \). * Order of \( R = (m \times n) \times (n \times m) = m \times m \) (Square matrix). We need to find matrices with the same order as \( A \), i.e., \( m \times n \). Let's check the product options: * PQ: \( P \) is \( m \times n \), \( Q \) is \( n \times n \). Product \( PQ \) order: \( (m \times n) \times (n \times n) = m \times n \). (Matches A). * QR: \( Q \) is \( n \times n \), \( R \) is \( m \times m \). Product defined only if \( n = m \), but matrices are non-square. Undefined or not matching. * RQ: \( R \) is \( m \times m \), \( Q \) is \( n \times n \). Undefined unless \( m=n \). * RP: \( R \) is \( m \times m \), \( P \) is \( m \times n \). Product \( RP \) order: \( (m \times m) \times (m \times n) = m \times n \). (Matches A). Therefore, \( PQ \) and \( RP \) have the same order as \( A \).
Step 4: Final Answer:

The matrices are \( PQ \) and \( RP \).