Question

If $A$ is $3\times 4$ matrix and $B$ is a matrix such that $A^{\prime }B$ and $B^{\prime }A$ are both defined, then the order of $B$ is

• A. $4\times 4$
• B. $3\times 3$
• C. $3\times 4$
• D. $4\times 3$

Answer 1

The correct option is C $3\times 4$

Given $A$ is a $3\times 4$ matrix and $B$ is a matrix such that $A^{\prime }B,B^{\prime }A$ are both defined.

We know that, we can multiply two matrices if, and only if, the number of columns in the first matrix equals the number of rows in the second matrix.

Since $A$ is a $3\times 4$ matrix, $A^{\prime }$ the transpose of $A$ is a $4\times 3$ matrix.

We have $A^{\prime }B$ is defined.

By the definition of multiplication of two matrices, we get the number of rows in matrix $B$ as $3$. $....\left(1\right)$

Also we have $B^{\prime }A$ is defined.

We have number of rows in matrix $B$ is $3$, then number of columns in matrix $B^{\prime }$ is $3$.

Since $A$ is a $3\times 4$ matrix and $B^{\prime }A$ is defined.

Therefore number of rows in matrix $B^{\prime }$ is $4$.

$\Rightarrow$ number of columns in matrix $B$ is $4$. $....\left(2\right)$

Thus by $\left(1\right)$ and $\left(2\right)$, we get

The order of matrix $B$ as $3\times 4$