Proof: Let A & B be Matrices; Show BA has a Row of Zeros

[-Dragoon-]

Homework Statement

28. 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 BA has a row of zeros.

Homework Equations

N/A

The Attempt at a Solution

$A = (aij)_{mxn}$ and $B = (bij)_{nxp}$. Assuming that the entries for jth column of A are all zeros, I infer that the jth column of BA is also a row of zeros. Let $cij$ be an entry in jth column. By the definition of multiplication:
$$cij = b_{i1}a_{j1} + b_{i2}a_{j2} + ...+ b_{in}a_{nj} = \sum_{k=1}^n b_{ik}a_{kj}$$

Since the jth column of A is zero, then there is:
$a_{1j} = a_{2j} = ... = a_{nj} = 0$. Hence, $cij = 0$ and, therefore, the jth column of BA is a column of zeros.

Did I do this proof correctly? Thanks in advance.

 

[Mark44]

Homework Statement

28. 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 BA has a row of zeros.

Homework Equations

N/A

The Attempt at a Solution

$A = (aij)_{mxn}$ and $B = (bij)_{nxp}$. Assuming that the entries for jth column of A are all zeros,

You're given that one row of A consists of zeros. You can't also assume that one column (column j) is all zeros.

I infer that the jth column of BA is also a row of zeros. Let $cij$ be an entry in jth column. By the definition of multiplication:
$$cij = b_{i1}a_{j1} + b_{i2}a_{j2} + ...+ b_{in}a_{nj} = \sum_{k=1}^n b_{ik}a_{kj}$$

Since the jth column of A is zero, then there is:
$a_{1j} = a_{2j} = ... = a_{nj} = 0$. Hence, $cij = 0$ and, therefore, the jth column of BA is a column of zeros.

Did I do this proof correctly? Thanks in advance.