Math 4133 - Linear Algebra
Exam 1 - 2016.02.17
YOUR NAME HERE
(remember Alt+7 gives a text bracket)

1. Consider the following system of equations:

({{{, ({{β x+5y-7z, = 1}, {-2x+4y+z, =2}, {3x+α y-2z, =4}})}})

(a) Solve this system of equations for {x,y,z} in terms of the parameters α and β.

(b) Convert the system to the form AX = B.

(c) Compute the determinant of A.

(d) For what value of β does this system of equations ALWAYS have a solution, independent of α?

(e) Assuming β is not the value found in part (b), express α as a function, f(β), for those values for which the det(A) = 0.

(f) Plot the function α = f(β) found in part (e).

(g) Let α = 2, and β be the value from part (d).  Plot all three planes together on the same graph.

(h) Set β to any value not found in part (d), and α = f(β).  Plot these three planes together on the same graph.

(i) Explain both graphs.

2. Consider the following system of equations AX=B:

(({{5, 6, -5, 3, 4, 2}, {4, -1, -2, 0, 3, 2}, {7, -9, 3, -2, -1, 1}, {2, -3, 5, 1, 1, 0}, {0, 2, 0, 5, 3, -2}, {-1, -2, -1, 2, 3, 4}}))(({{(Subscript[x, 1])}, {(Subscript[x, 2])}, {(Subscript[x, 3])}, {(Subscript[x, 4])}, {(Subscript[x, 5])}, {(Subscript[x, 6])}})) = (({{1}, {0}, {-1}, {0}, {1}, {0}}))

(a) Compute det(A).

(b) Row reduce (A|(Subscript[I, 6])) to compute (A^(-1)).

(c) Use Cramer’s Rule solve for the variable (Subscript[x, 4]).

3. Convert the following matrix to upper triangular form using only left multiplication by elementary matrices of type III.  Also, store the corresponding values from the type III matrices correctly into a lower triangular matrix L so that A=LU.  Verify your results.

A = (({{1, -3, 2}, {2, 4, 0}, {-2, 3, 3}}))