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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 $BA=I$?
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