Using Matlab programming

1. By typing A = round (10*rand(5)) and B = round (8*rand(5)), you will generate 2 random 5x5 matrices with integer entries. Use these 2 matrices and MATLAB commands to compute the following matrices.

(a) AB (b) BA

(c) (AB)-1 (d) (A)-1(B)-1

(e) (B)-1(A)-1 (f) (A - B)(A + B)

(g) A2 - B2 (h) A2 + AB - BA - B2

(i) (A - 3I)5 (j) (A + B)-1

(k) A-1 + B-1

(l) Determine which, if any of the matrices are equal.

2. Find the solution(s) of the following linear systems. Use the MATLAB command rref to immediately find the reduced row echelon form of a matrix. Enter the coefficient matrix A and vector b. When entering the vector b, type b=[b1 b2 b3]'. The ' indicates transpose and changes b from a row vector to a column vector. Type rref([A b]). This is asking for the reduced row echelon form of the augmented matrix [A b]. For each system write by hand all solutions (or state that there is no solution) on your printout near the problem.

(a)

2x1	+	x2	+	3x3	+	2x4	=	-4
-x1			-	x3	+	x4	=	2
3x1	-	2x2	+	x3	+	4x4	=	-21
4x1	+	7x2	+	5x3	+	4x4	=	12

(b)

2x1

+

6x2

-

3x3

-

x4

+

18x5

+

2x6

=

-103

x2

+

3x3

+

4x4

+

12x5

+

9x6

=

-61

-2x1

+

3x2

-

5x5

+

6x6

=

63

-x1

+

7x2

+

8x3

+

12x4

+

39x5

+

x6

=

-208

7x1

+

4x2

-

3x3

+

5x4

+

66x5

=

-469

(c)

2x1	+	6x2	-	3x3	=	-1
		x2	+	x3	=	4
-2x1	+	3x2			=	0
-x1	+	7x2	+	8x3	=	12
7x1	+	4x2	-	3x3	=	5

(d)

x1	-	x2	+	x3	-	x4	=	0
x1	+	2x2	-	3x3	+	4x4	=	0
3x1	-	x2	+	2x3	+	4x4	=	0
	-	x2	-	x3	+	2x4	=	0

3. Determine the polynomial of degree 4 or less that passes through these 5 points, by using the rref command to solve an appropriate linear system. (See example 2 on page 26 of our text.) Yours will be a 4^th degree polynomial in this form: . On your printout write the polynomial and the linear system you used to find it.