Quiz 1

Problem 1: A linear system has the following coefficient matrix:

Factor , where L is a lower triangular matrix and U is an upper triangular matrix. You need not show the steps.

Write L as a product of elementary matrices. Find the inverse for each of the elementary matrices and use them to compute L^-1.

Problem 2: A matrix A is defined over Z₃, (has entries from the field Z₃)

Compute A^-1 by find the row-reduced echelon form of the augmented matrix (A | I). Note: Your arithmetic needs to be done in Z₃. See preliminaries for review.

R1+R2 ~ R2

R2 + R3 ~ R3

2R3

2R2

R3 + R2 ~ R2

2R2 + R1 ~ R1

Problem 3: Given the matrix equation A² + 2A - I = 0. Find A^-1.
Hint: Solve for I.

Therefore

Problem 4: Find the equation for the parabola in the form that passes through the three points P, Q, and R on the plane whose x-y coordinates are P(1,4), Q(-1,6) and R(2,9).

Solution: Plug the x and y from each point into P(x).

Let

Problem 5: Let and be a
polynomial. P(A) is defined as .
Compute P(A).

Problem 6: The equation of a plane in R³ has the form where a, b, c, d are constants. Find the equation of the plane through the points P(1,1,2), Q(1,2,0) and R(2,1,5).

Choose d = 1, the a = 3, b = -2 and c = -1.

Problem 7: The seven stage model for a loggerhead turtle population is given by the matrix equation

where is the initial population vector for each of the seven stages, is the Leslie matrix describing the changes in population for each stage over a years time, k is the number of years from the initial time and is the population vector k years after the initial starting time.

Let

if is the initial turtle population vector, what will the population be after 10 years, 50 years according to this model?

Solution:

Problem 8: Let A be a matrix whose block structure is given by

where

Find A^-1.

Solution: Using Mathcad. Build the matrix A from the blocks.