Matrix Algebra and the Algebra of Numbers

- A comparative study: continued -

Linear Algebra Project

Objective:

This is the continuation of the previous lab with the same title. You will discover more subtle similarities and differences between the algebra of numbers and the algebra of matrices.

1. Experimenting with matrices that act like 0's and 1's.

> with(linalg):

a. Solving the equation M X = M

In the algebra of numbers the equation has exactly one solution, , if m is not zero.

Is X = I the only solution?

Let us look at the matrix M:

> M:=matrix(3,3,[[1,-1,2],[2,1,-1],[4,-1,3]]);

, the identity matrix, is certainly a solution of .

Since M is not zero must I be the only solution?

The answer is: NO! Here is another solution:

> X:= matrix(3,3,[[2,-2,3],[-5,11,-15],[-3,6,-8]]);

You can check this by finding

> multiply(M,X);

More solutions for MX = M

Note that if X is a solution of the equation , then so is .

You can verify this by taking the matrix X given in the preceding section, squaring it and using Maple to see if it is indeed a solution.

In fact any power of X will be a solution.

Working it out:

Raise X to the power 2. This is how Maple does powers:

> X2:=evalm(X^2);

Now let us check to see if X2 is a solution of .

> multiply(M,X2);

Let us recall the matrix M :

> evalm(M);

Indeed X2 is a solution of the equation.

How many solutions does the equation have?

Find another solution.

Find, if possible, two solutions for the matrix equation, , where R is

> R:= matrix(3,3,[[1,1,2],[1,0,1],[0,1,1]]);

Solution:

is actually a system of three linear equations, and the Gauss-Jordan method of our first lab may be used to solve them by reducing the augmented matrix ( ) to its echelon form:

> augment(R,R);

> gaussjord(R,R);

This shows that the system has infinitely many solutions. Here is one of them:

> X:= matrix(3,3,[[2,-1,3],[1,0,3],[-1,1,-2]]);

Now we verify that this is a solution of :

> multiply(R,X);

This is our matrix R given above. (You will have to press enter again in front of R again before entering the multiply(R,X) command, so that R is in the memory again. Otherwise you will get an error message.)

b. Can the product of two non-zero matrices be zero?

If the product of two numbers is zero, then at least one of them has to be zero. What about matrices?

Problems:

If , then . Use matrices M and ( X-I ) from part a. to do the multiplication.

What do you get?

What do you expect the product to be equal to? Try it.

A paper and pencil exercise: For the matrix V given below, find a non-zero matrix Y so that their product is zero.

V:= matrix(2,2,[[1, 2],[-1, 1]]).

What do you conclude?

c. Does Zero have cube roots?

If , what is N ?

You say N must be 0. That certainly is true. In fact if N were a real number that would be the only solution. What about matrices?

Find powers of N, where

> N:= matrix(3,3,[[-22,-3,-25],[40,6,44],[16,3,16]]);

Find and . (To find M to the power n in Maple write: evalm(M^n);)

What do you notice? What will you get for ?

Use paper and pencil to find a 2 by 2 non-zero matrix such that its square is the zero matrix.

d. Two more abnormalities to experiment with:

d1. If AB=AC , what do you conclude?

If a, b and c are non-zero numbers and , we would conclude that . Now we are going to take three matrices and test this algebraic result for them:

> A:=matrix(2,2,[[1,2],[3,6]]);

> B:=matrix(2,2,[[1,2],[-1,3]]);

> C:=matrix(2,2,[[-9,6],[4,1]]);

Compute AB and AC .

When you compare the results, you will see that this example shows that another multiplicative property of the real numbers does not carry over to matrices. Which property of real numbers is it?

Check to see if and are equal for the matrices given below.

d2. What is special about this matrix?

> Q := matrix([[-744, -496, -1736], [-300, -200, -700], [405, 270, 945]]);

If you discover its secret, then use pencil and paper to create a 2 by 2 matrix with the same property.

2. Experimenting with the transpose of a matrix

Maple uses "transpose(A)" to denote , the transpose of A.

Do you think is equal to ?

The following A, B and C are imported from the previous lab:

> A:= matrix(4,4,[[1,0,-1,3],[2,1,4,-2],[0,-5,0,1],[-1,2,-1,3]]);

> B:= matrix(4,4,[[2,1,0,7],[-2,5,1,2],[4,1,3,-6],[-1,1,-8,2]]);

> C:= matrix(4,4,[[2,1,-5,-2],[-3,4,-1,3],[0,2,0,1],[7,13,-52,3]]);

Finding and

> At:= transpose(A);

> Bt:= transpose(B);

Now let us check to see if is equal to . Let us name the matrix as ABt .

> ABt:=transpose(multiply(A,B));

> AtBt:= multiply(At,Bt);

The two matrices are not the same.

Discovering a relationship between and its components and

Lab assignment: Use Maple to experiment with different ways you can multiply the above matrices. See if you can come up with a formula that connects to some combination of and .

If you have discovered a formula, bring matrix C into play to test it.

If your formula still works, then generate some random matrices to verify it. If it still works, there is a very high probability that your formula is true. However, in mathematics the truth can only be established by a rigorous proof. If that challenges you, then try to prove your formula or look it up in a linear algebra book.

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A Writing Assignment

On the basis of your observations and results gained in the previous projects, write a paper titled "A Comparative Study of the Algebra of Numbers and Matrices."

Do not assume that your reader has studied linear algebra, but has mathematical maturity on the level of calculus.

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MSIP Grant #120A80089-98: "Three Urban Calculus Reform Programs: Adopting the Best" 1998-2001