Question

Let both $AB'$ and $B'A$ be defined for matrices $A$ and $B$. If the order of $A$ is $n \times m$, then the order of $B$ is:

• A. $n \times n$
• B. $n \times m$
• C. $m \times m$
• D. $m \times n$

Hint:

For matrix multiplication to be valid, ensure the number of columns in the first matrix equals the number of rows in the second matrix.

Answer 1

The Correct Option is D

To establish the dimensions of matrix $B$, we must adhere to the rules of matrix multiplication:

• For the product $AB'$ to be valid, the number of columns in matrix $A$ must be equal to the number of rows in the transpose of $B$ ($B'$). Since the number of rows in $B'$ corresponds to the number of columns in $B$, matrix $B$ must possess $n$ columns.
• If $B$ is an $m \times n$ matrix, its transpose $B'$ will have dimensions $n \times m$.
• For the product $B'A$ to be defined, the number of columns in $B'$ must match the number of rows in $A$. If $A$ is $n \times m$, then $B'$ must be $m \times m$. This implies $B$ must be $m \times n$ for $B'A$ to be defined. Considering both $AB'$ and $B'A$, the requirement for $B$ is that it must be $m \times n$.

Consequently, for both products $AB'$ and $B'A$ to be defined, the dimensions of $B$ must be $m \times n$. The correct dimensions for $B$ are thus $m \times n$.