MATH 203 MIDTERM I
Please work out each of the given problems without the use of a calculator. Credit will be based on the steps that you show towards the final answer. Show your work.
Consider the matrix
Find a 2x1 matrix v such that Av = 2v.
gives us the two equations
v1 + v2 =
-2v1 + 4v2 =2v2
-v1 + v2
-2v1 + 2v2 =
the second equation is a multiple of the first. We can pick
Please work out each of the given problems. Credit will be based on the steps that you show towards the final answer. Show your work.
Answer the following true or false and
explain your reasoning.
the determinant of both sides of the equation
(det A)(det A) =
(det A)2 = 1
and the result follows.
–2 1 2] T represent a sample of a function
of four equispaced points.
Determine the final average and detail coefficients by computing.
To compress the data, we set any detail coefficient equal to zero that is below 1 in absolute value. In this case, we set the last detail coefficient equal to zero. The compressed data is
Next we use the inverse transformation to decompress the data. We compute
In the city of Digraphville, there are four
food-processing plants: the apple plant, the beet plant, the carrot plant, and the
dairy plant. There are one-way
roads from the apple plant to the beet plant and to the dairy plant. There is also a one-way road from the beet plant to the
carrot plant. There are two-way
roads from the apple plant to the carrot plant, from the beet plant to the dairy
plant and from the carrot plant to the dairy plant.
Sketch the digraph for this situation
Use the adjacency matrix to determine how many ways are there to drive
from the apple plant to the dairy plant using no more than four roads counted
The number of ways to drive from the apple plant to the dairy plant using no more than four roads counted with multiplicity is given by
[A + A2 + A3 + A4]14
Hence there are 16 ways of getting from the apple plant to the dairy plant using no more than four roads.
Prove that if A,
B, and C are
n x n matrices, then
A(B + C)
= AB + AC
[A(B + C)]ij = S(aik(B + C)kj) = S(aik(bkj + ckj) )
= S(aik(bkj + ckj) ) = S(aikbkj + aikckj) = Saikbkj + Saikckj
= (AB)ij + (AC)ij
Prove that if
w are solutions to the matrix equation Ax
b and if r + s = 0, then
rv + sw
is a solution to the homogeneous equation Ax = 0.
Since v and w are solutions to the matrix equation equation. We have
Av = Aw = b
A(rv + sw) = A(rv) + A(sw)
= rAv + sAw = rb + sb = (r + s)b = (0)b = 0
down one thing that your instructor can do to make the class better and one
thing that you want to remain the same in the class.
(Any constructive remark will be worth full credit.)