1Given a system of $n$ linear equations in $n$ variables $Ax = b$, which of the following is a necessary and sufficient condition for Cramer's rule to be applicable to solve for the unique variables?
2Calculate the volume of the parallelepiped determined by the vectors defined by the points $P_1(1, 1, 0)$, $P_2(1, 0, 1)$, $P_3(0, 1, 1)$, and $P_4(1, 1, 1)$, where the vectors are $\vec{P_1P_2}$, $\vec{P_1P_3}$, and $\vec{P_1P_4}$.
3Which of the following sets of vectors form a basis for $\mathbb{R}^3$?
4Find the equation of the plane that contains the point $P(-2, 1, 0)$ and is parallel to the plane $-8x + 6y - z = 4$.
5Let $v = (1, 2, -4)$ and $u = (-2, 0, 1)$. Find the vector $w_1$ which is the orthogonal projection of $v$ onto $u$.
6If $\lambda$ is an eigenvalue of an invertible matrix $A$, which of the following statements must be true?
7For which values of $a$ is the matrix $A = \begin{pmatrix} a & 2a \\ 2 & 2a+2 \\ \end{pmatrix}$ invertible?
8Determine the distance from point $B(2, 2, -3)$ to the plane defined by $x - y + 3z = 1$.
9Given a $3 \times 6$ matrix $A$ in reduced row echelon form with 3 pivot columns, which of the following is true about the solution space of $AX = 0$?
10If $M$ is an invertible $3 \times 3$ matrix and $MC = B$, where $B$ is a $3 \times 2$ matrix, how is matrix $C$ calculated?
