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Find the order of B matrix if A is a $3 \times 4$ matrix and B is a matrix such that A’B and BA’ are both defined.:
$(a){\text{ 3}} \times {\text{4}}$
$(b){\text{ 3}} \times 3$
$(c){\text{ 4}} \times {\text{4}}$
$(d){\text{ 4}} \times 3$

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Hint: In the above given question, we are asked to find out the order of the matrix for which various conditions are already provided. By using these conditions one by one, first of all find out the number of rows in the matrix and then using the other conditions, find out the number of columns in the given matrix.

Complete step-by-step answer:

We are given in the question that, A is a $3 \times 4$ matrix, therefore we know that,
Order of A’ will be ${\text{4}} \times 3$.
Since, it is given that A’B exists and we already know that A’ is a ${\text{4}} \times 3$ matrix.
Hence, we can conclude that B must have 3 rows.
Also, it is given in the question that, BA’ exists, therefore we come to know that,
B must have 4 columns.
Now, we know that B has 3 rows and 4 columns, so we can say that B is a $3 \times 4$ matrix.
Hence, the required solution is the option $(a){\text{ 3}} \times {\text{4}}$.

Note: When we face such types of problems, the key concept is to have a good knowledge of the matrices and its properties particularly regarding order of the matrix. By using these properties and the conditions already provided to us in the question, we can obtain the required solution.
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