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 B is of the type(A)$3 \times 4$(B)$3 \times 3$(C)$4 \times 4$(D)$4 \times 3$

Hint: Find the order of${A^t}$. Use the fact that ${A^t}B$ is defined as $\Rightarrow$ number of columns of ${A^t}$ = the number of rows of B. Also, $B{A^t}$ is defined$\Rightarrow$ the number of columns of B = the number of rows of ${A^t}$
This will give us the required answer.

Complete step by step solution: We are given that A is a $3 \times 4$ matrix. Note that$3 \times 4$ is the order of matrix A.
Also, we have a matrix B such that ${A^t}B$ and $B{A^t}$ are both defined.
We are asked to find the order of B.
${X^t}$denotes the transpose of a matrix X and is a matrix obtained by changing the rows of matrix X into columns and the columns into rows.
Therefore, if $m \times n$ is the order of matrix X, then the order of ${X^t}$ will be $n \times m$.
Now, for a matrix Y, we say that the product ${X^t}Y$ is defined if and only if the order of Y is $m \times r$.
That is, the number of columns of the matrix ${X^t}$ should be equal to the number of rows of matrix Y.
Now, we have the order of matrix A which is $3 \times 4$. This means that the order of ${A^t}$ is $4 \times 3$.
As we are given that ${A^t}B$ is defined, we can conclude that the order of B is of the form$3 \times s$where s is some positive integer. This is because the number of columns of ${A^t}$= the number of rows of B.
Now,$B{A^t}$is defined$\Rightarrow$the number of columns of B = the number of rows of ${A^t}$
No. of columns of B = s and number of rows of ${A^t}$= 4.
$\Rightarrow s = 4$
That is, the number of columns of B = 4.
We already know that the number of rows of B = 3.

Hence B is a $3 \times 4$ matrix.

Note: 1) The order of a matrix, also called as dimension of a matrix, is the number of rows and columns of the matrix and is given by the expression $m \times n$ where m denotes the number of rows in the matrix and n denotes the number of columns in the matrix.
2) If A is an$m \times n$matrix and B is an $n \times r$matrix, then the order of matrix AB is $n \times r$.