MTH 177 Graded Assignment 3 (modules 5-6)
Unless specified otherwise, please use SciLab to do any matrix computations where possible. Be sure to
incorporate your SciLab code into your solutions; that's your supporting work, and if you have an
incorrect answer I can see if it's as simple as a typo. I'd prefer that you not print the screen directly from
SciLab, but take the time to copy and paste the relevant bits into a document, and format everything
neatly, so that I know (1) what the question is, (2) what the work is, and (3) what the solution is.
Problem 1:
For the matrix
 0 1
 1 3
A 
4 2

10 7
4 10 
2 3

5 7 

3 2 
(a) Write At
(b) State whether or not A is symmetric
(c) Compute tr ( A)
Problem 2:
For the complex valued matrices
3
6  5i 
 4

A  3 3  2i 5  3i 


6  5i 5  3i
1 
3i
3  2i 
 4

B  3i
7 2  7i 


3  2i 2  7i
2 
 7 3  2i 4  i 
C   3  2i
0
6 


 4  i
6
2 
(a) Find the conjugate transpose of each matrix. (Can be done easily by hand or with SciLab)
(b) State which of the matrices are Hermitian.
(c) Use SciLab to compute the quantity ABC 2 . This would be extremely messy to do by hand, and
the point of the question is to check that you know the syntax for entering complex valued
matrices!
Problem 3:
Prove that for an m n matrices A and B , if A and B are symmetric, then A  B is symmetric.
This is a very short proof, and you don't need to use elements - just use the properties of the transpose.
In particular, what does ( A  B)T  ?.
Then, construct a counterexample that shows that the converse does not hold (show that you can find
two matrices A and B where A  B is symmetric...but A and B individually are not).
Problem 4:
By hand (show work), using Gauss-Jordan elimination, find the inverse of the matrix A shown below.
Don’t forget about the row operation calculator at Linear Algebra Toolkit! Check your solution using the
inv(A) command in SciLab.
 1 1 2 
A   5 1 5 


 2 3 0 
Problem 5:
a) Verify that for a 2  2 matrix
a b 
A

c d 
the inverse can be computed using the formula
A1 
1  d b 
ad  bc  c a 
The way to do this (since the proposed candidate for A1 is already given) is to verify that AA1  I , and
that A1 A  I . Note you do have to verify the multiplication works in both orders, since matrix
multiplication is not in general commutative.
b) What does this formula tell you about the condition required on A in order for it to be
invertible?
c) Apply the above to give the inverse of the matrix A given below, and state the values of x and
y that make A non-invertible.
0 
y 2
A 
x  3
 y
Problem 6 (quick SciLab question):
One of the properties of matrix inverses is that ( AB ) 1 is NOT equal to A1 B 1 , but instead
( AB)1  B 1 A1 ; the inverse of a product is the product of the inverses in reverse order. Invent a
couple of 3  3 matrices, and verify this fact for them: use SciLab to compute ( AB ) 1 , A1 B 1 , B 1 A1 ,
and see for yourself which two of those three quantities are in fact equal.
Problem 7 (another SciLab question – don’t do any row reducing by hand):
For the system of equations
2 x1  x2  3 x3  x4
 x1  x2  2 x3  x4
x1  x3  x4
3 x1  2 x2  6 x4
 7
 2
 10

1
(a) Express as a matrix equation AX  B .
(b) Find A1 .
(c) Compute the solution to the system from X  A1 B .
(d) And check by solving by row reducing the augmented matrix for the system.
Problem 8:
Send me an encrypted message!
(a) Invent a message. Don't tell me what it is. It should consist of a comprehensible English phrase
or sentence, however.
(b) Invent a 3  3 square matrix for the encryption. Don't tell me what that is either. Do check and
make sure that matrix is invertible before you use it.
(c) Encrypt your message using the substitution A  1 , B  2 etc (as discussed in the cryptography
section), and further encrypt it using your matrix.
(d) The only things I want from you are (1) the encrypted message, transmitted as a string of
numbers, and (2) the key to decrypting your message. Consult the cryptography lecture if you
don't recall what that key is.
Do use SciLab for the calculations. I'll decrypt your message based on what you've sent me. As long as
it's recognizable words, it'll mean you did it right.
For this question, no matter how you submit the rest of your assignment, you MUST send me
(electronically, i.e by email) a TEXT DOCUMENT into which you have
1) TYPED the string of numbers representing your message. Or copy and paste, but rearrange the
form – the message should be transmitted as a string, not a matrix – one thing I’m looking for
is do you know how you’re supposed to interpret the matrix.
2) COPIED and PASTED the matrix which is the key. This, take verbatim from SciLab and don’t
modify anything.
Do not hand transcribe the numbers – I need to be able to copy/paste back to SciLab to decode, and I
have no wish to be typing a gazillion six digit decimals.