If any two matrices A and B of suitable orders then

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Dimension Property of Matrix:

One property that is unique to matrices is the dimension property.

This property has two parts:

Part 1: The product of two matrices will be defined if the number of columns in the first matrix is equal to the number of rows in the second matrix.

Part 2: If the product is defined, the resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Explanation:

Let A be a matrix of order [a × b]    ----(i)

And B be a matrix of order [b × d]   ----(ii)

Since we have to find the multiplication of A and B 

So we let the column of A = Row of B  [which is b] 

From the properties of multiplication 

The order of AB will be [a × d] 

But, According to the question, the order of AB is [n × n]

i.e The value of a and d should be n

i.e a = d = n     ----(iii)

Now, From (i), (ii), and (iii), we get

The order of A = [n × b]

The order of B = [b × n]

Here the order of A and B are not the same 

But it can be the same when b = n

Here, We can say that to satisfy the above question the order of two matrices need not be the same.

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It is said that a matrix $A$ is said to be invertible if there exists a matrix $B$ such that $AB=I=BA$, but if we only know $AB=I$, can we be sure that $BA=I$?