Could someone explain all parts on how to solve this question and please explain
ID: 3404474 • Letter: C
Question
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.Explanation / Answer
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.