(B) (5 pts.) Fill in the blanks. These sentences that you fill in the blanks must be completely correct.
(B.1) If M is 5 × N nonzero matrix, then rank(M) + nullity(M) = ....
(B.2) If $Ax = 0$ has only the trivial solution, then the rank of A is the number of .... of A.
(B.3) If A is square matrix and $Ax = b \ne 0$ has a unique solution, then .... cannot be an eigenvalue of A.
(B.4) If dim(V) = n, then any subset of V containing more than n vectors is linearly ....
(B.5) If the nonzero n x n matrix B is singular, then rank(B) = ....
Q2) (15 pts.)
(a) Let $A = \begin{bmatrix} 1 & 4 \\ 2 & N \\ -1 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & N & 0 & -1 \\ -1 & 3 & 1 & 6 \\ 1 & 1 & -3 & 0 \end{bmatrix}$. Find the product $A^T B$.
(b) Which of the following matrices are in reduced row echelon form?
$A = \begin{bmatrix} 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 0 & 2 & 3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}$, $C = \begin{bmatrix} 1 & 2 & 0 & 4 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 1 \end{bmatrix}$, $D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$, $E = \begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 1 & 0 & 0 & N \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}$
Write your answer in the box provided:
(c) Find the rank of the matrix $M = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 5 \\ 3 & 4 & 7 \\ 4 & 5 & 9 \end{bmatrix}$.
