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 \(A\)