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 determine the order of matrix $B$, we need to consider the conditions for matrix multiplication:

• Matrix multiplication $AB'$ means $A$ is multiplied by the transpose of $B$. For this multiplication to be defined, the number of columns in $A$ must equal the number of rows in $B'$. Since $B'$ (the transpose of $B$) has the number of rows equal to the number of columns in $B$, $B$ must have $n$ columns.
• The matrix $B'$ resulting from the transpose of $B$ will have dimensions $n \times m$ if $B$ is $m \times n$.
• For $B'A$ multiplication, the number of columns in $B'$ must match the number of rows in $A$. If $A$ is $n \times m$, then $B'$ must be $m \times m$ for the multiplication $B'A$ to be defined. However, since $A$ is $n \times m$, it suffices to consider only the requirement that $B$ has dimensions $m \times n$ to satisfy both $AB'$ and $B'A$ being defined.

Conclusively, the order of $B$ must be $m \times n$ for both products $AB'$ and $B'A$ to be defined. Therefore, the correct answer is $m \times n$.