Problems in class

1. Is $(A+B)^2 = A^2 + 2 AB + B^2?$ If not, correct the equation.

2. Find the matrix $E$ such that all the conditions below are satisfied for every $3\times 3$ matrix $A$,
   • row two of $EA$ is (2 times row 1 of $A$ + row 2 of $A$),
   • the first and third rows of $EA$ equal the corresponding rows of $A$.

3. Consider the plane of all linear combinations of the vectors $\begin{bmatrix} 1\\2\\3 \end{bmatrix}$ and $\begin{bmatrix} 3\\2\\1 \end{bmatrix}$, where the three coordinates are interpreted as the $x-$, $y-$ and $z-$ coordinates. How do we check if all points on the $x-$axis lie in the plane?

4. Problem in text. For each row below, choose the only $3\times 3$ matrix $B$ such that for all matrices $A$ the condition given in that row (answers may be different for different rows)
   • $$BA = 4A$$
   • $$BA = 4B$$
   • $BA$ has rows 1 and 3 of $A$ reversed, and row 2 unchanged
   • All rows of $BA$ are the same as row 1 of $A$