Problems in class

1. Is \((A+B)^2 = A^2 + 2 AB + B^2?\) If not, write the correct rule.

2. Let \(A\) be a \(3\times 3\) matrix. Find a matrix \(E\) such that
   • 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. Choose the only \(3\times 3\) matrix \(B\) such that for all matrices \(A\),
   • \[BA = 4A\]
   • \[BA = 4B\]
   • \(BA\) has rows 1 and 3 of \(A\) reversed, and row 2 unchanged
   • All rows of \(BA\) are the same as row 1 of \(A\)