MATH 203 MIDTERM I

Part 1

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. 

 

Problem 1

Consider the matrix

  1. 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

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     


  2. Find the adjoint of A  

    Solution

    We first find the cofactors.  We recall that we can find the cofactor Aij by finding the determinant of its minor and multiplying by (-1)i+j.  We have

                A11  =  15         A12  =  -10       A13  =  3

                A21  =  -5         A22  =  4           A23  =  -1

                A31  =  -12       A32  =  8           A33  =  -2

    The adjoint is the transpose of the matrix of cofactors. 

               

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     


  3. Use the adjoint formula and parts A. and B. to find the inverse of A.  

    Solution

    We have that

                  adj A
                A-1  =                   
                              det A

    Hence

               

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Problem 2

Let

  1. Find A2 and A3.  

    Solution

    We have

                 

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     


  2. Make a conjecture about Ak.  

    Solution

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     


  3. 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.  

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     


 

Problem 3 

Let 

Find a 2x1 matrix v such that Av = 2v.

Solution

We write

           

or

           

this gives us the two equations

            v1 + v2  =  2v1

            -2v1 + 4v2 =2v2

or

            -v1 + v2  =  0

            -2v1 + 2v2  =  0

Notice the second equation is a multiple of the first.  We can pick

            v1  =  1      v2  =  1

or

               

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 





Part 2

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. 

 

Problem 4 

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.

Solution


False, for example, let

             

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


B.     If A is a matrix with A2  =  In then either det(A)  =  1 or det(A)  =  -1.

Solution

True.  Take the determinant of both sides of the equation

            det A2  =  det In

            (det A)(det A)  =  1

            (det A)2 =  1

and the result follows.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Problem 5 

Let v  =  [1   2   1   2] T represent a sample of a function of four equispaced points. 

A.     Determine the final average and detail coefficients by computing.  A2A1v.  

Solution

We have

               

so that

                

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

B.     Using a threshold of e  =  1 determine the compressed data and then compute the wavelet.

Solution

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

       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Problem 6 

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  

Solution

       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


B.     Write down the adjacency matrix  

Solution

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.  

Solution  

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 

We have

       

Hence there are 16 ways of getting from the apple plant to the dairy plant using no more than four roads.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Problem 7 

Prove that if A, B, and C are n x n matrices, then

A(B + C)  =  AB + AC

Solution

Proof

We have 

        [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 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Problem 8 

Prove that if v and 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.  

Solution

  Proof

Since v and w are solutions to the matrix equation equation.  We have

        Av  =  Aw  =  b

We have

        A(rv + sw)  =  A(rv) + A(sw)

        =  rAv + sAw  =  rb + sb  =  (r + s)b  =  (0)b  =  0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

Extra Credit:  Write 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.)