Question

If \( A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \) and \( B = \begin{bmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{bmatrix} \), then which one of the following is True?

• A. \( A^T B B^T A = B^T A A^T B \)
• B. The orders of \( A^T B B^T A \) and \( B^T A A^T B \) are equal
• C. The orders of \( A + B, A^T B, B A^T \) are equal
• D. Rank of \( A \) and \( B \) are equal

Hint:

To find the order of a product of matrices, say \( M_{p \times q} N_{r \times s} \), the number of columns of the first matrix must be equal to the number of rows of the second matrix (i.e., \( q = r \)). The resulting matrix will have an order of \( p \times s \). Keep track of the orders of the matrices at each step of the multiplication to determine the order of the final product.

Answer 1

The Correct Option is B

The matrix \( A \) has order \( 2 \times 3 \).
The transpose of \( A \), \( A^T \), has order \( 3 \times 2 \).
The matrix \( B \) has order \( 2 \times 3 \).
The transpose of \( B \), \( B^T \), has order \( 3 \times 2 \).
Let's find the order of \( A^T B B^T A \): \( A^T \) ( \( 3 \times 2 \) ) \( B \) ( \( 2 \times 3 \) ) results in a matrix of order \( 3 \times 3 \).
\( (A^T B) \) ( \( 3 \times 3 \) ) \( B^T \) ( \( 3 \times 2 \) ) results in a matrix of order \( 3 \times 2 \).
\( (A^T B B^T) \) ( \( 3 \times 2 \) ) \( A \) ( \( 2 \times 3 \) ) results in a matrix of order \( 3 \times 3 \).
So, the order of \( A^T B B^T A \) is \( 3 \times 3 \).
Now let's find the order of \( B^T A A^T B \): \( B^T \) ( \( 3 \times 2 \) ) \( A \) ( \( 2 \times 3 \) ) results in a matrix of order \( 3 \times 3 \).
\( (B^T A) \) ( \( 3 \times 3 \) ) \( A^T \) ( \( 3 \times 2 \) ) results in a matrix of order \( 3 \times 2 \).
\( (B^T A A^T) \) ( \( 3 \times 2 \) ) \( B \) ( \( 2 \times 3 \) ) results in a matrix of order \( 3 \times 3 \).
So, the order of \( B^T A A^T B \) is \( 3 \times 3 \).
Since both \( A^T B B^T A \) and \( B^T A A^T B \) have order \( 3 \times 3 \), their orders are equal.
Let's examine the other options:
Option (A) \( A^T B B^T A = B^T A A^T B \): Matrix multiplication is not generally commutative, so this equality does not necessarily hold.

Option (C) The orders of \( A + B, A^T B, B A^T \) are equal: \( A + B \) is defined and has order \( 2 \times 3 \).
\( A^T B \) has order \( 3 \times 3 \).
\( B A^T \) has order \( 2 \times 2 \).
The orders are \( 2 \times 3 \), \( 3 \times 3 \), and \( 2 \times 2 \), which are not equal.

Option (D) Rank of \( A \) and \( B \) are equal: Two matrices of the same order do not necessarily have the same rank.
The ranks depend on the linear independence of their rows or columns.
Therefore, the only true statement is that the orders of \( A^T B B^T A \) and \( B^T A A^T B \) are equal.