Math 260 Basic Linear Algebra Fall 2001 Midterm Exam II

1  2  3  4  TOTAL

Name : Student Number: Section: Duration: 100 minutes

(15 pts.) Q1. Consider the matrices

A = ( \begin{matrix}1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 41 & 2 \\ 0 & 0 & 1 & 0 \end{matrix} ) , B = ( \begin{matrix}97 & 98 & 98 & 99 \\ 79 & 1 & 0 & 7 \\ −1 & 0 & 0 & 2 \\ 1 & 1 & 1 & 1 \end{matrix} ) , C = ( \begin{matrix}6 & 5 & 4 & 3 & 2 & 1 \\ 6 & 5 & 4 & 3 & 2 & 0 \\ 6 & 5 & 4 & 3 & 0 & 0 \\ 3 & 2 & 1 & 0 & 0 & 0 \\ 3 & 2 & 0 & 0 & 0 & 0 \\ 3 & 0 & 0 & 0 & 0 & 0 \end{matrix} ) .

Compute the following determinants and write your answers next to the “ = ” signs.

a) det A = e) det(AB) =

b) det B = f) det(A + A) =

c) det C = g) det(3B) =

d) det(A^{−1}) = h) det(B^{T}) = (20 pts.) Q2. Determine the value(s) of a for which the following system has (a) no solution,

(b) a unique solution, (c) infinitely many solutions. In cases (b) and (c), write down the solution(s).

\begin{matrix}2x & − & y & + & 2az & + & u & = & −1 \\ −2x & + & ay & + & (3 − 4a)z & & & = & 11 \\ 2x & − & y & + & (2a + 1)z & + & (1 − a)u & = & 1 \\ −2x & + & y & + & (1 − 2a)z & − & u & = & 4a + 3 \end{matrix} \}

(20 pts.) Q3. Let A = ( \begin{matrix}1 & 1 & 0 & 0 \\ 2 & 0 & 0 & −2 \\ −1 & 0 & 1 & 1 \\ 2 & −1 & −1 & −2 \end{matrix} ) .

a) Obtain A^{−1} using Gauss-Jordan elimimation method.

b) Compute the volume of the four-dimensional parallelepiped determined by the vectors (1, 2, −1, 2), (1, 0, 0, −1), (0, 0, 1, −1), (0, −2, 1, −2).

c) Find det(A^{−1}).

d) Find the cofactor matrix cof (A). (Hint: You may use the results of previous calculations.)

(20 pts.) Q4. Let T : ℝ^{4} → ℝ^{⊯} be given by T (x, y, z, w) = (x + y, −z + w, z). a) Show that T is linear,

b) Given the ordered bases B_{1} = {(1, 1, 1, 1), (1, 0, 0, 0), (1, 0, 1, 0), (0, 0, 0, 1)} and B_{2} = {(−1, 0, 0), (0, −1, 1), (4, 0, 2)} for ℝ^{⋭} and ℝ^{⊯} respectively, find the matrix representation of T with respect to B_{1} and B_{2}.

c) Find a basis for the null space N(T).

d) Find dim N(T) and the rank of T.