Question

If A and B are two matrices such that A + B and AB are both defined, then
A . A and B are two matrices not necessarily of the same order
B. A and B are square matrices of the same order
C. Number of columns of A is equal to the number of rows of B
D. None of these

Answer 1

Hint: Here in this question, A and B are two matrices in which addition and multiplication are both defined. For this, we suppose two matrices of the same order then by using the addition and the multiplication property, we find out the option which follows our answer.

Complete step by step Solution:
Given A and B are two matrices that A + B and Ab are both defined.
Let A be a matrix of order $({{a}_{1}}\times {{a}_{2}})$ and B be a matrix of order $({{b}_{1}}\times {{b}_{2}})$
As A.B is defined
The number of columns in A = number of rows in B
Then ${{a}_{2}}={{b}_{1}}$………………………………… (1)
Since A + B is defined
The number of rows in A = number of rows in B
And the number of columns in A = number of columns in B
Then ${{a}_{1}}={{b}_{1}}$ and ${{a}_{2}}={{b}_{2}}$……………………………..(2)
From (1) and (2), we have,
${{a}_{1}}={{a}_{2}}={{b}_{1}}={{b}_{2}}=n$
So A is of order $n\times n$ and B is of order $n\times n$
We know matrices that have the same number of rows and the same number of columns as square matrices.
So A and B are square matrices of the same order.

Therefore, the correct option is (B).

Note: We must remember that the addition of matrices is possible only when the number of rows is equal to the number of columns of the given matrix. Also, multiplication matrices are of the same order.
For example: the matrices $B(4\times 3)$$A(4\times 4)$and $B(4\times 4)$are possible for both the operations.
If we take $A(4\times 3)$ and $B(3\times 4)$, multiplication is possible but the addition is not possible.
Similarly in matrices $A(4\times 3)$and in $B(4\times 3)$,addition is possible but multiplication is not possible.