Could someone explain all parts on how to solve this questio

Could someone explain all parts on how to solve this question and please explain it in as much detail as possible and do all the parts 1-3 of the questions. This is Linear alegebra

74. Guided Proof Prove that if the product AB is a square matrix, then the product BA is defined Getting Started: To prove that the product BA is defined, you need to show that the number of columns of B equals the number of rows of A (i) Begin your proof by noting that the number of columns of A equals the number of rows of B. (ii) Then assume that A has size m x n and B has size n X p. (iii) Use the hypothesis that the product AB is a Square matrix.

Solution

Numerical example to start with :

Let A be a 3x5 matrix. Let B be a pxq matrix.

For the product AB to make sense, p must be equal to 5.

If this is satisfied , then AB is a 3xq matrix.

Given that AB is a square matrix.

This forces q to be 3.

Thus A is 3x5 and B is 5x3.

Hence BA also makes sense (as the number of columns of B is 3=the number of rows of A). In fact BA is also a square matrix of order 5.

The argument above holds for any A (mxn) and B(pxq).

AB well defined implies p=n and AB square implies q=m.

So the number of columns of B =m matches the number of rows of A =m.

Hence BA is well defined and in fact a square matrix of order m.