(a) Show that if A is an m x n matrix and A(BA) is defined, then B is an n x m matrix.

Question

(b) Show that if A has a row of zeros and B is any matrix for which AB is defined, then AB also has a row of zeros.

(c) Find a similar result involving a column of zeros.

Thanks!!!

Answer 1

(a) Let A be an m x n matrix and B be an n x m matrix. Then A(BA) is defined because the number of columns in A is equal to the number of rows in B.

(b) Let A be an m x n matrix with a row of zeros and B be an n x p matrix. Then AB is defined because the number of columns in A is equal to the number of rows in B. Since A has a row of zeros, the product AB will also have a row of zeros.

(c) Let A be an m x n matrix with a column of zeros and B be an n x p matrix. Then AB is defined because the number of columns in A is equal to the number of rows in B. Since A has a column of zeros, the product AB will also have a column of zeros.