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
- Use the definition of the determinant to
find |A|.
**Solution**Notice that the even permutations on three elements are (1,2,3) (2,3,1) (3,1,2) and the odd permutations are (1,3,2) (2,1,3) (3,2,1) The even permutations correspond respectively to the terms 0 + 0 + 0 = 0 and the odd permutations correspond respectively to the terms 0 + 10 + (-12) = -2 Taking the odds minus the evens gives det|A| = 0 - (-2) = 2 - Find the adjoint of A
**Solution**We first find the cofactors. We recall that we can find the cofactor A _{ij}by finding the determinant of its minor and multiplying by (-1)^{i+j}. We haveA _{11}= 15 A_{12}= -10 A_{13}= 3A _{21}= -5 A_{22}= 4 A_{23}= -1A _{31}= -12 A_{32}= 8 A_{33}= -2The adjoint is the transpose of the matrix of cofactors. - Use the adjoint formula and parts A.
and B. to find the inverse of A.
**Solution**We have that adj A A^{-1}= det AHence
Let
- Find A
^{2}and A^{3}.**Solution**We have - Make a conjecture about A
^{k}.**Solution** - Use induction to prove your conjecture
from part B.
**Solution**For the case k = 1, this is trivial Assume the statement is true for k. Then We need to show that conjecture is true for k + 1. We have so by mathematical induction, the theorem is true.
Let
Find a 2x1 matrix v such that Av = 2v.
We
write
or
this
gives us the two equations
v
-2v or
-v
-2v Notice
the second equation is a multiple of the first. We can pick
v or
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. A.
If A and B
are n x n matrices and AB
= 0, then either A
= 0 or B = 0.
B.
If A is a matrix with A
True. Take
the determinant of both sides of the equation det
A
(det A)(det A) =
1
(det A) and the result follows. Let v
= [1
–2 1 2] A.
Determine the final average and detail coefficients by computing.
A
We have
so that
B.
Using a threshold of e =
1 determine the compressed data
and then compute the wavelet.
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. A.
Sketch the digraph for this situation
B.
Write down the adjacency matrix
We have
C.
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
with multiplicity.
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 + A We have
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
We have
[A(B + C)]
= S(a
= (AB)
Prove that if
rv + sw
is a solution to the homogeneous equation Ax = 0.
Since A We have A(r = rA
(Any constructive remark will be worth full credit.) |