This is from the book Elementary Linear Algebra (the 9th edition) by Bernard Kol

Question

This is from the book Elementary Linear Algebra (the 9th edition) by Bernard Kolman. Section 1.3, #28.

Part (a) Let A be an m x n matrix with a row consisting entirely of zeros. Show that if B is an n x p matrix, then AB has a row of zeros.

Part (b) Let A be an m x n matrix with a column consisting entirely of zeros and let B be p x m. Show that BA has a column of zeros.

For both of these, we have to write a short proof, and I really don't know where to begin or how to write the proof. It makes sense to me that there would be the row or column of zeros, but I don't know how to write a general proof.

Explanation / Answer

For part a)
Begin with a matrix A that has a row of 0s. So use for instance a m x n, that equals a 2 x 2 (i.e. 2 rows, 2 columns) Note that In order to multiply two matrices, the first matrix should have the same number of rows as the columns in the second matrix. This should be an assumption.

A = [A B
0 0]

and B is a matrix n x p, that we will make a 2 x 2

B = [C D
E F]

Multiplying these matrices gives an AB matrix of size 2 x 2, where each corresponding value is multiplied (i.e. 1st row, 1st column value of each matrix, so for A = A, for B = C, given an AB matrix for 1st row, 1st column answer of AC]

AB = [AC BD
(0*E)=0 (0*F)=0]

AB = [AC BD
0 0]

This should satisfy the proof. You would need to write step by step and explain the given, the steps, and the conlusion. As with what was done with part a) you would do with part b) but make the column in matrix a all 0s as opposed to the row.

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