Linear algebra deals with arithmetic operations that are performed on rectangular arrays (tables) of numbers. The algebraic properties of the basic operations on real numbers can be summarized by saying:

(1) a list of numbers can be added in any order;

(2) a list of numbers can be multiplied in any order;

(3) adding 0 to a number doesn't change its value;

(4) multiplying a number by 1 doesn't change its value;

(5) adding a number and its opposite yields 0;

(6) multiplying a number by its reciprocal yields 1;

(7) when we multiply a number by a sum of numbers, we get the same result as when we

multiply the number by each number in the sum and then add the products;

(8) numbers are partitioned into three categories: positive, zero (there is only one zero),

and negative, with the properties we are familiar with (the fact that multiplying two

negative numbers yields a positive number is the strange one).

There is only one restriction: there is no reciprocal of 0.

Addition and its opposite.

Matrices come in different sizes. The size of a matrix is described by two numbers, n x m, where the number of rows is n and the number of columns is m. Only matrices that are the same size (with the same number of rows and the same number of columns) can be added.

Example of adding two 3 x 2 matrices. (We will use the notation of the TI-83/84 calculator.)

[[1 2] [[-1 3] [[0 5]

[3 4] + [-2 0] = [1 4]

[5 6]] [0 2]] [5 8]]

What is the zero matrix among 3 x 2 matrices?

[[0 0] [[1 2] [[1 2]

[0 0] + [3 4] = [3 4]

[0 0]] [5 6]] [5 6]]

A zero matrix is an array of 0's. There are many zero matrices, one for each size.

Each matrix has its opposite.

[[1 2] [[-1 -2] [[0 0]

[3 4] + [-3 -4] = [0 0]

[5 6]] [-5 -6]] [0 0]]

We can only add matrices of the same size, but the properties of addition are the same as for real numbers. Conditions (1), (3), and (5) hold.

Multiplication and reciprocals.

(For this part of the lesson, students should already use a TI-83/84 calculator. This can be their first introduction to the matrix menu.)

The situation here is different. Two matrices can be multiplied only when the number of columns of the first matrix is equal to the number of rows of the second matrix. The resulting product is a matrix which has the number of rows of the first matrix and the number of columns of the second one.

So, for example, a 3 x 5 matrix may be multiplied by a 5 by 2 matrix, and the result will be 3 x 2 matrix.

Task.

Using the matrix editor, create a 3 x 5 matrix [A] and a 5 x 2 matrix [B], and compute [A]*[B] from the home screen.

Notice that a matrix M whose size is n x m can be multiplied by itself only if n = m. Such matrices are called square matrices, for obvious reasons, and they play an important role in matrix multiplication.

Remark.

The "row by column" rule of multiplying matrices should be shown to students because it explains which matrices can be multiplied by each other, and which cannot. But practice with calculating matrices by hand should not be given.

If we multiply three matrices A, B, and C, having sizes n x m, m x k, and k x l, then

(A * B) * C = A * (B * C),

and the result is an n x l matrix. So we can change the order in which the operations are performed. But we cannot change the order of the matrices themselves.

In general A * B ≠ B * A, even when both products are defined. So property (2) of multiplication of numbers given above only partly generalizes.

The concept of one, as a unit of multiplication, generalizes as follows:

A square matrix, n x n, which has 1's on the diagonal that go from the top left to the bottom right, and zeros everywhere else, plays the role of 1.

Example.

[[1 2] [[1 0] [[1 2]

[3 4] * [0 1]] = [3 4]

[5 6]] [5 6]]

and

[[1 0 0] [[1 2] [[1 2]

[0 1 0] * [3 4] = [3 4]

[0 0 1]] [5 6]] [5 6]]

Only square matrices can have reciprocals, but not every square matrix has a reciprocal. Computing a reciprocal is complex, and students should only do it using calculators. Property (6), multiplying a number by its reciprocal yields 1, holds for reciprocals.

Task.

Use the matrix editor to enter a few 3 x 3 matrices, and try to compute their reciprocals using the reciprocal key from the home screen.

When we combine addition and multiplication of matrices, property (7) above holds.

Finally, the concept of positive, zero, and negative numbers has a corresponding concept for square matrices. Every square matrix M has a number associated with it that is called its determinant, Det(M). A square matrix M has an inverse if and only if Det(M) ≠ 0. Also the determinant of a product of two square matrices is the product of their determinants, Det(M*N) = Det(M)*Det(N).

Final remarks.

This analysis shows that a student with a good knowledge of arithmetic and a rudimentary knowledge of algebra can learn how to use matrices rather easily, providing that he/she has access to a calculator or computer that makes the actual computation easy.

Before the availability of modern technology, matrices were taught as a part of algebra,

and not as an extension of arithmetic, because the arithmetic aspects were too tedious for hand calculation. But this limitation is no longer valid.

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