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

Let A be a matrix of order mxn and B be a matrix of order nxp.

Then the order of the product AB by definition is mxp.

Given AB is a square matrix, hence m must be equal to p.

So A has order mxn and B has order nxm.

Thus the product BA is clearly defined.(No of rows of A=No of columns of B = m)