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Matlab and Numerical Approximation
ASSIGNMENT 1
1. Enter the matrices
and carry out the following:
(a) Verify that (A + B) + C = A + (B + C).
(b) Verify that (AB)C = A(BC).
(c) Verify that A(B + C) = AB + AC.
(d) Decide whether AB is equal to BA.
(e) Find (A + B)^{2}, (A^{2} + 2AB + B^{2}) and
(A^{2} + AB + BA + B^{2}).
(f) Find A^{2}  B^{2}, (A  B)(A + B) and
(A^{2} + AB  BA + B^{2}).
2. Enter
and do the following:
(a) Compute A^{2}, A^{3}, etc. Can you say what A^{n} will be? Explain why this is true.
(b) Compute B^{2}, Can you explain why this is true. What does this tell you about
matrix multiplication that is different from squaring numbers ?
(c) Find AB and BA. What do you learn from this that is not true for
multiplication
of numbers? (hint: if a is a real number and a ^{2} = 0, then a = 0).
3. Find the inverse of the matrices (if they exist) and check that the result is
correct by
multiplying the matrix times its inverse.
4. Generate an 8 × 8 matrix and an 8 × 1 vector with integer entries by
A = round(10 * rand(8)); b= round(10 * rand(8; 1));
(a) Use °ops to count the number of floating point operations needed to solve Ax
= b
using the \ notation.
(b) Reset °ops to zero and resolve the system using the row reduced echleon form
of
the augmented matrix [A b] (i.e., U = rref([A b])). The last column of U (call
it y)
is the solution to the system Ax = b. Count the °ops needed to obtain this
result.
(c) Which method was more efficient?
(d) The solutions x and y appear to be the same but if we look at more digits we
see
that this is not the case. At the command prompt type format long . Now look at
x and y, e.g., type [x y]. Another way to see this is to type x  y.
(e) which method is more accurate ? To see the answer compute the socalled
residuals,
r = b  Ax and s = b  Ay. Which is smaller?
When you are finished reset format to short  format short.
5. Given the matrices
(a) AX + B = C,
(b) AX + B = X,
(c) XA + B = C,
(d) XA + C = X.
6. Let A = round(10 * rand(6)). Change the sixth column as follows. Set
B=A' % (take the transpose of $A$)
now type
A(:,6)= sum (B(1:5,:))'
Can you explain what this last command does? Compute
det(A)
rref(A)
rank(A)
Can you explain why A is singular?
7. Let A = round(10*rand(5)) and B = round(10*rand(5)). Compare the following
pairs
of numbers.
(a) det(A) and det(A').
(b) det(A + B) and det(A) + det(B).
(c) det(AB) and det(A) det(B).
(d) det(A^{1}) and 1/ det(A).
8. Look at help on magic and then compute det(magic(n)) for n = 3;,4, 5,
…,10. What
seems to be happening? Check n = 24 and 25 to see if the patterns still holds.
By pattern
I mean try to describe in words what seems to be happening to these
determinants.
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