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This is the Exam of Matrix Algebra which includes Unique Solution, System, Solutions, Infinitely Many Solutions, General Solution, Parametric Vector, Determinants, Matrices, Traffic Flow etc. Key important points are: Subspace, Depend, Linear, Matrix, Inverse, Dimension, Conditions, Vector, Spanned, Three Vectors
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Math 221, Section 203, Winter 2008 Page 1 of 20
April 21, 2008, 15:30–18:
No books. No notes. No calculators. No electronic devices of any kind.
1 2 3 4 5 6 7 8 9 10 total/
This exam has 10 problems. The first 9 problems are common to all three sections, the last problem is section-specific.
Problem 1. (5 points) Solve the following linear system. Your answer will depend on k.
x 1 + x 2 − 2 x 3 + x 4 = 1 2 x 1 + 2x 2 − 3 x 3 + x 4 = 2 3 x 1 + 3x 2 − 4 x 3 + x 4 = k
Problem 2. (5 points) Let the matrix A be
A =
0 1 0 t 0 3 1 t 1 0 − 1 t
Find the inverse of A if possible. Your answer will depend on t.
Problem 4. (5 points) Let L : R^2 → R^2 defined by L
([x y
[ (^) x y − x
. Let T =
be a basis of R^2. (a) Find the representation of L with respect to the basis T. (b) Let v be a vector in R^2 such that [L(v)]T =
. Find v.
Problem 5. (6 points) Let L : R^3 → R^3 be a linear transformation for which we know that
L
(a) What is L
(b) Find the matrix A of L with respect to the standard basis. (c) Find det(A).
Problem 7. (5 points) Compute the determinant of the matrix
1 0 t 0 1 0 0 1 0 2 3 0
Problem 9. (9 points)
Suppose ~vn =
xn yn zn
(^) is the state vector of a dynamical system at time n. Suppose
the time dependence of the dynamical system is given by the equations
xn+1 = xn − yn + zn yn+1 = −xn + 3yn − zn zn+1 = −xn + 3yn − zn (a) Find all state vectors that do not change in time (these are vectors such that ~vn+1 = ~vn, for all n). (b) Given that ~v 0 =
, find ~v 100.
(c) Given that ~v 1 =
(^) find all possible values for ~v 0.
Problem 10. (9 points) Let q(x, y, z) = − 3 x^2 − 8 xz + 5y^2 + 3z^2 be a quadratic form.
(a) Find a symmetric matrix A such that q(x, y, z) = [x y z]A
x y z
(b) The eigenvalues of A are 5 and -5. Find an orthogonal matrix P and a diagonal matrix D such that D = P −^1 AP. (c) Find a change of variables
x y z
x′ y′ z′
(^) and a quadratic form q′(x′, y′, z′) such that q′^ is equivalent to q and q′^ does not involve any cross-product term.
