MTH 285 Graded Assignment 2 (modules 4-6)
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: Let
 3 x 2 
A
2
 5 4x 
x  2
x2 


B 1
2 x  1
 0
5 x3 

(a) Compute the product BA
(b) Is it possible to compute AB ? Explain.
Problem 2:
Use a typical element proof to prove the property shown below (distribution of scalar over matrix
addition) (assume sizes are such that addition is defined):
c( A  B )  cA  cB
Problem 3:
We know that in general, matrix multiplication is not commutative ( AB  BA ). However, it is possible
to have specific matrices which commute, and certain classes for matrices for which multiplication is
commutative. In particular, if A and B are diagonal matrices, then AB  BA . Prove it.
Recall that a diagonal matrix is a square matrix where all entries not on the diagonal are zero. The way
to set this up and get it moving notationally is “Let A be an n  n diagonal matrix. So ai. j  0 when
i  j .”, do the same thing to define B , and make use of the multiplication formula.
A hint is to expand out the sum – it should look like this when i  j
n
xi , j   ai ,qbq , j  ai ,1b1, j  ai ,2b2, j  ...  ai ,i bi , j  ...  ai , j b j , j  ...  ai ,nbn, j
q 1
And will look a little different when i  j (what does this turn into for xi ,i ?)
That’s what you need to think about - what has to happen when i  j , and when i  j ? [For
convenience, you can ignore the i  j case; it’s no different than the i  j case with the terms slightly
re-ordered.)
Note that for Problems 2 and 3, these must be general proofs - you may not work out the proof with a
matrix of a specific size (say, using a 3  3 to show the calculations). Typical element proofs, and you'll
need to work with the (sigma notation) definition of matrix-matrix multiplication in Problem 3.
Problem 4:
For the complex valued matrices
3 6  5i 
 4

A   3
6 5  3i 
5  5i 5  3i
1 
3
3  2i 
 4i

B 3
7 2  7i 
3  2i 2  7i
2 
 7 5i 3  2i 
C   5i
0
6 
3  2i 6
2 
(a) Find the conjugate transpose of each matrix.
(b) State which of the matrices are Hermitian.
(c) Use SciLab to compute the quantity ABC 2
Problem 5:
The zero factor property of real numbers states that if xy  0 , than either x  0 or y  0 (this is the
property that allows you to solve quadratic equations by setting them equal to zero and factoring).
a) This property does not always hold for matrices – construct a counterexample; i.e. give two
matrices A and B (with sizes compatible for matrix multiplication), where AB  0 (the zero
matrix), but neither A nor B is a zero matrix.
b) However, it does hold when A is a certain type of matrix – when is it true that AB  0 implies
B  0 ? Once you figure it out, prove it. The proof is instantaneous, once you realize what A has
to be! Hint: I’m looking for a broad category of matrix – one true-but-not-what-I’m-looking-for
answer I sometimes get is “the identity” – while, yes, it’s certainly true that if IB  0 , then
B  0 , that’s too specific. Think of a property A has to have…
Problem 6 (SciLab question – don’t do any row reducing by hand):
For the system of equations
x1  6 x2  3x3  x4
 x2  2 x3  x4
x1  2 x2  x3  x4
 5
 0
 10
2 x2  6 x4

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 7:
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 
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.
1
A
y
x  3
0 
Problem 8:
Factor the matrix
 1 1 0
A   3 4 1 
 2 3 6 
into a product of upper and lower triangular matrices: A  LU .