Math 315

Linear Algebra and Differential Equations

Homework Hints

Assignment 1

Determine how many unknown real numbers are necessary to specify a symmetric n by n matrix.

A hint is that

1 + 2 + 3 + ... + k = k(k+1)/2

Determine whether the following statement is true or false and give your reasoning.

Every diagonal matrix is a multiple of the identity matrix.

Use MATLAB to enter the following matrix and vector.

A = [ 1 0 -1 ; 2 5 3 ; 5 -1 0];
b = [1; 1; -2];

x = A\b

Find an integer for the entry in the 2nd row, 2nd column of A so that the solution x = A\b is not defined.

If you like trial and error, the answer is between -4 and 4. If you prefer to think, realize that for there not to be a solution, it must be the case that the second equation is inconsistent with the first and third equations. Try writing the equations out using a variable c instead of the coefficient 5. Try solving for x, y, and z in terms of c. There is a c for which there is no solution.

Suppose that substances S₁, S₂, S₃, and S₄ contain the given percentages of vitamins A, B, C, and F by weight.

Vitamin	S1	S2	S3	S4
A	25%	19%	20%	3%
B	2%	14%	2%	14%
C	8%	4%	1%	0%
F	25%	31%	25%	16%

Mix the substances so that the resulting mixture contains precisely 3.85 g of vitamin A, 2.30 g of vitamin B, 0.80 g of vitamin C, and 5.95 g of vitamin F. How many grams of each substance have to be contained in the mixture? Discuss what happens when we require that the resulting mixture contains 2.00 g of vitamin B instead of 2.3 g.

Hint: Let x₁, x₂, x₃, x₄ be the grams of the substances. Write down four equations which calculate the grams of each vitamin in the mixture. Enter the corresponding coefficient matrix and vector into MATLAB to solve the system of equations.

Use MATLAB to determine graphically the geometry of the set of solutions to the system of equations

  x + 3y + 4z = 5
 2x +  y +  z = 1
-4x + 3y + 5z = 7

Hint: After setting up the graphics display, you can use the command view([0,1,0]) to get a better view of the solution point.