Answer :

We are given that,

Order of matrix A = 2 × 3


ATB and BAT are defined matrices.


We need to find the order of matrix B.


We know that the transpose of a matrix is a new matrix whose rows are the columns of the original.


So, if the number of rows in matrix A = 2


And, number of columns in matrix A = 3


Then, the number of rows in matrix AT = number of columns in matrix A = 3


Number of columns in matrix AT = number of rows in matrix A = 2


So,


Order of matrix AT can be written as


Order of matrix AT = 3 × 2


Thus, we have


Number of rows of AT = 3 …(i)


Number of columns of AT = 2 …(ii)


If ATB is defined, that is, it exists, then


Number of columns in AT = Number of rows in B


2 = Number of rows in B [from (ii)]


Or,


Number of rows in B = 2 …(iii)


If BAT is defined, that is, it exists, then


Number of columns in B = Number of rows in AT


Substituting value of number of rows in AT from (i),


Number of columns in B = 3 …(iv)


From (iii) and (iv),


Order of B = Number of rows × Number of columns


Order of B = 2 × 3


Thus, order of B is 2 × 3.


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