Question

If $A$ is $3 \times 4$ matrix and $b$ is the matrix such that $A'B, BA'$ are both defined then $B$ matrix is of type
A. $3 \times 4$
B. $3 \times 3$
C. $4 \times 4$
D. $4 \times 3$

Answer 1

Let us assume that $B$ is $m \times n$ matrix and $A'$ and $B'$ are defined as the transpose of $A,B$ respectively. So if $A$ is $3 \times 4$ matrix then its transpose will be $4 \times 3$ matrix. $A'B$ is defined so the number of columns of $A'$ must be equal to the number of rows of $B$ and similarly for $BA'$. So we will find $m \times n$ and we can get our answer.

Complete step by step solution:
Now firstly we need to find the meaning of the transpose matrix. Let we have the matrix $A$ of order $i \times j$ then for $A'$ to be the transpose of $A$ it will become$j \times i$.
For example: WE have the matrix A as \left[ { \begin{array}{*{20}{c}} a&b;&c; \end{array} \\\ \begin{array}{*{20}{c}} d&e;&f; \end{array} } \right] of order $2 \times 3$
Then $A'$ is defined as the transpose of A which is A' = \left[ {\begin{array}{*{20}{c}} a \\\ b \\\ c \end{array}{\text{ }}\begin{array}{*{20}{c}} d \\\ e \\\ f \end{array}} \right]
Which is $3 \times 2$ matrix
For the multiplication of the matrices A and B which means to get $AB$ number of columns in $A$ must be equal to the number of rows in $B$
That means if A is $i \times j$ then for AB to be defined B must be $j \times k$ matrix and here $j$ is the column of A and row of B.
And we are given that matrix $A$is $3 \times 4$ matrix and transpose of $A$ which is $A' =$ $4 \times 3$
Also we assumed that $B$ be $m \times n$ matrix then the transpose of $B$ which is $B' = n \times m$
Now for $A'B$ to be defined number of columns of $A'$ must be equal to number of rows of $B$
So number of columns in $A'$ is $3$
Number of rows in $B$ is m
So $m = 3$
For $BA'$ to be defined the number of columns of $B$ must be equal to the number of rows of $A'$
So $n = 4$
Now we assumed that $B$ is $m \times n$ matrix

Therefore $3 \times 4$ is the answer.

Note:
If matrix $A$ is given $m \times n$ and matrix $B$ is given by $P \times Q$ then for AB to be defined
$n = P$ and for BA to be defined $m = Q$
So for both AB and BA to be defined $n = P, Q = m$
Therefore both must have the same number of rows and columns.