3. The following matrix is the augmented matrix of a linear system. Find all values of k that
make the system consistent with infinitely many solutions.
$\begin{bmatrix} k & 1 & : & 2 \\ 6 & -1 & : & -2 \end{bmatrix}$
(A) 6 (B) -6 (C) $\frac{2}{3}$ (D) all real numbers (E) none of the given options
4. If A is a 5 x 1 matrix, B is a 5 x 2 matrix, C is a 1 x 4 matrix, D is a 5 x 4 matrix and E
is a 4 x 5 matrix. What can you say about the size of ($B^TD + DE)^T$?
(A) 2 x 4 (B) 1 x 4 (C) undefined 23 (D) 4 x 2 (E) none of the given options.
5. If A = $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, Find all 2 x 2-matrices B that satisfy AB = BA.
(A) B = $\begin{bmatrix} y & -x \\ y & y \end{bmatrix}$ such that x and y are real numbers
(B) B = $\begin{bmatrix} y & y \\ 0 & y \end{bmatrix}$ such that y is a real number
(C) B = $\begin{bmatrix} y & x \\ 0 & y \end{bmatrix}$ such that x and y are real numbers
(D) B is either $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ or $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
(E) none of the given options.
