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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).
and do the following:
(a) Compute A2, A3, etc. Can you say what An will be? Explain why this is true.
3. Find the inverse of the matrices (if they exist) and check that the result is
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));
(b) Reset °ops to zero and resolve the system using the row reduced echleon form
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
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
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 $A$)
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
(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,
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.