If Row 1 = Row 3 of B, does Row 1 = Row 3 of AB? [Strang P76, 2.4.7(b)] 
True or False: If $\color{  #009900}{\text{rows $1$ and $3$ of $B$ are the same}}$, so are rows $1$ and $3$ of $AB.$
  Answer: False

Attempt: Since we're given information only about the rows of $B$, the key is to regard $B$ as a matrix of row vectors. So I must express the matrix multiplication of $AB$ as  column $\cdot$ row.
Then $AB = \left[\vec{A_1} \quad \vec{A_2} \quad \vec{A_3}\right]_{3 \times 3} \left[\begin{matrix} 
\color{  #009900}{\vec{B_1} = \vec{B_3}}  \\
\vec{B_2} \\
\color{  #009900}{\vec{B_3}}
\end{matrix}\right]_{n \times p} 
:=  \vec{A_1}\color{  #009900}{\vec{B_3}} + \vec{A_2}\vec{B_2} +  \vec{A_3}\vec{B_3}.$
How would you then calculate/determine the answer?
Supplementary thanks to the answer by T. Bongers:
I don't perceive why one would "guess" this as false; the analogous claim for columns is true: 
$\text{From P76, 2.4.7.(a): If columns $i$ and $j$ of $B$ are the same, then so are columns $i$ and $j$ of $AB$.}$ 

If it's false, you just need one counterexample. Consider two matrices
$$A = \left(\begin{array}{c} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right)$$
and 
$$B = \left(\begin{array}{c} 1 & 1 & 1 \\ 2 & 3 & 4 \\ 1 & 1 & 1 \end{array}\right)$$
Then the first entry in $AB$'s first row is $1 + 4 + 3 = 8$, while the first entry in $AB$'s third row is $7 + 16 + 9 = 32$.
