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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: Null Space, Matrix, Value, Basis, Contained, Span, Determinant, Invertible, Coordinate Vector, Standard Matrix
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Math 221, Section 101, Fall 2007 Page 1 of 17
December 18, 2007, 15:30โ18:
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. (6 points) Find a basis for the null space of the matrix
2 โ 4 1 t 1 โ 2 2 t 1 โ 2 1 2 t 1 โ 2 1 t
Your answer will depend on the value of t.
Problem 2. (4 points) Find the inverse of the matrix
Problem 3. (6 points) Find a 2 ร 2 matrix A such that ( 1 2 0 1
Problem 5. (6 points) (a) (4 points) Find the determinant of the matrix
1 4 10 t
(b) (2 points) For what values of t is B invertible?
Problem 6. (6 points)
(a) Explain why B = {
} forms a basis for R^2.
(b) Find the coordinate vector of
in the basis B.
(c) Suppose the standard matrix of a linear transformation T : R^2 โ R^2 is ( 2 โ 3 0 2
Find the matrix of T with respect to the basis B, i.e., find [T ]B.
Problem 7. (8 points) Consider the matrix
A =
(a) (2 points) Verify that 1 is an eigenvalue of A. (b) (3 points) Find all eigenvalues of A. (c) (3 points) For each eigenvalue, find the dimension of the corresponding eigenspace.
Problem 9. (8 points) On a remote planet, moisture is present in clouds, on the continents, and in the seas. Each year 80% of the cloud moisture falls onto the land and 10% falls into the seas. Each year 15% of the land moisture evaporates directly into the clouds and 65% runs into the seas. Each year 70% of the sea moisture evaporates into the clouds. Assume we know also that the total amount of water or moisture on the planet is 86 trillion litres. (a) Draw a diagram of the moisture transfer on this planet. (1 point) (b) Write a system of linear equations describing the moisture transfer using the variables c, l, and s for water content in clouds, on land, and in the seas, respectively. Find the transition matrix of this dynamical system. (2 points) (c) Suppose the planets moisture distribution is in equilibrium. What is the an- nual precipitation (volume of water falling onto land from the clouds)? ( points) (d) A large meteor falls onto the planet, causing all sea water to evaporate into the clouds. (The impact has no other effect on the moisture distribution, but it does cause mass extinction of aquatic species.) Write down the matrix which represents the change in the state vector
( (^) c sl
caused by the impact. (2 points)
Problem 10. (8 points) Consider the quadratic form
Q(x 1 , x 2 , x 3 ) = 9x^21 + 7x^22 + 11x^23 โ 8 x 1 x 2 + 8x 1 x 3.
(a) If B is the 3 ร 3 matrix representing Q, then 15 is an eigenvalue of B. Find the remaining eigenvalues. (2 points) (b) Find a change of variables that will remove the cross terms from Q. (6 points)
