Answer

Verified

405k+ views

**Hint:**To solve this question, we have to know the condition for the product of two matrices to exist. The product of 2 matrices A, B which is A$\times $B is defined when the number of columns of the first matrix which is A is equal to the number of rows of the second matrix which is B. Using this property, we can get the rows and columns of the matrix B as required in the question.

**Complete step-by-step solution:**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 $y, the transpose A’ will be of the order y$\times $x.

Let us consider the first product in the question which is AB’. The order of B’ will be l$\times $k. For AB’ 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 B’A. The order of B’ will be l$\times $k. For B’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.

**Note:**In the answer, we got the inference that the order of matrix A and B are the same for the condition that AB’ and B’A be defined. So we can infer from this result that for two matrices C and D, the products CD and DC are defined when the conditions D is of the order of transpose of C and vice versa which is C is of the order of transpose of D should be satisfied.

Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

Assertion The resistivity of a semiconductor increases class 13 physics CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

What is the stopping potential when the metal with class 12 physics JEE_Main

The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main

Using the following information to help you answer class 12 chemistry CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE

What organs are located on the left side of your body class 11 biology CBSE