Matlab and Numerical Approximation


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, (A2 + 2AB + B2) and
(A2 + AB + BA + B2).
(f) Find A2 - B2, (A - B)(A + B) and
(A2 + AB - BA + B2).

2. Enter

and do the following:

(a) Compute A2, A3, etc. Can you say what An will be? Explain why this is true.

(b) Compute B2, 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 a2 = 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 so-called 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

solve the matrix equations:

(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


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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