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Question

Answers

A. $m\times m$

B. $n\times n$

C. $n\times m$

D. $m\times n$

Answer
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In the notation of order of a matrix M of order m$\times $n, m is the number of rows and n is the number of columns. Let C, D be two matrices of orders $a\times b$ and $x\times y$ respectively.

For the product C$\times $D to be defined, we have to apply the condition for the product to be defined which is the number of columns of C is equal to the number of rows of D. Mathematically it is b = x$\to \left( 1 \right)$.

The important point that we have to note here is that there is no condition for the number of rows of first matrix C and the number of columns of second matrix D.

In the question, it is given that the matrix A is ordered m$\times $n. Let the order of B be k$\times $l.

For a matrix A of order x$\times $y, the transpose Aâ€™ will be of the order y$\times $x.

Let us consider the first product in the question which is ABâ€™. The order of Bâ€™ will be l$\times $k. For ABâ€™ to be defined, from the equation-1, we get that

$\therefore $ l = n$\to \left( 2 \right)$

Let us consider the first product in the question which is Bâ€™A. The order of Bâ€™ will be l$\times $k. For Bâ€™A to be defined, from the equation-1, we get that

$\therefore $ k = m$\to \left( 3 \right)$

From equations- 2 and 3 we get the order of B as m$\times $n.

$\therefore $ The order of matrix B is m$\times $n. The answer is option D.

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