Question

If A is a $3\times 4$ matrix and B is a matrix such that ${{A}^{T}}B$ and $B{{A}^{T}}$ are both defined, then the order of B is.
(a) $3\times 4$
(b) $3\times 3$
(c) $4\times 4$
(d) $4\times 3$

Answer 1

Hint: In this question, we need to use the condition that multiplication of matrices is possible only when the number of columns in the first matrix is equal to the number of rows in the second matrix. Then by using the given conditions we can get the number of rows and columns in the matrix B.

Complete step by step answer:
A matrix is a rectangular arrangement of numbers.
In the matrix representation $m\times n$ m denotes the number of rows in the given matrix and n denotes the number of columns in the given matrix.
Two matrices A and B can be multiplied if the number of columns in the matrix A is equal to the number of rows in the matrix B.
TRANSPOSE OF A MATRIX
Let A be a matrix of order $m\times n$. Then the $n\times m$ matrix obtained by interchanging the rows and columns of A is called transpose of A.
As given the order of the matrix A as $3\times 4$ we can get the order of the matrix ${{A}^{T}}$
Now, the order of the matrix ${{A}^{T}}$ will be $4\times 3$.
Let us assume the order of the matrix B as $m\times n$.
Now, as given in the question that ${{A}^{T}}B$ is defined. So, we can say that the number of columns in the matrix ${{A}^{T}}$will be equal to the number of rows in the matrix B.
Now, from this condition as the number of columns in the matrix ${{A}^{T}}$are 3 we get,
$\therefore m=3$
Now, as given in the question that $B{{A}^{T}}$ is defined. So, we can say that the number of columns in the matrix B will be equal to the number of rows in the matrix ${{A}^{T}}$.
Now, from this condition as the number of rows in the matrix ${{A}^{T}}$are 4 we get,
$\therefore n=4$
Thus, the order of the matrix B is $3\times 4$
Hence, the correct option is (a).

Note: It is important to note that the number of columns and rows of the matrix B should be equated to the corresponding rows and columns of the matrix ${{A}^{T}}$ according to the given condition but not to the number of rows and columns of the matrix A.
Because by taking the transpose of a certain matrix the number of rows and columns gets interchanged. So, if we equate it correspondingly according to the conditions then the order of the matrix B changes.