Matrix Product and Identitites

The scene:

Spy headquarters at Matrix World. Jill Bonza introduces Maximus Bond to Trinity.

OH WOW Trinity, I loved you in the movie "Matrix", its just so cool to meet you. Jill says you can tell me all about the identities here.

Well Max, I can tell you about matrix identities, but first you need to know about multiplying matrices together. Let me know when you're familiar with that Max.

See Matrix basics if you need to brush up background for this.

MATRIX MULTIPLICATION or FINDING THE PRODUCT OF MATRICES

How do we multiply a matrix by a matrix? The definition is not the natural thing one would first think of, but, strange as it at first seems, it is what has been found to be very useful.

DEFINITION

Given two matrices A = (a_ij) and B = (b_ij) we can only find AB if the number of columns of A is the same as the number of rows of B.

Suppose that A is mxn and B is nxp, (i.e. the number of columns of A is the same as the number of rows of B), then C = AB is an mxp matrix where the elements of C, c_ij, are given by

c_ij = a_i1b_1j + a_i2b_2j + ...+ a_inb_nj

i.e. To get the ij element of C, multiply the elements of row i from A with the corresponding elements of column j from B and add.

EXAMPLE. Find AB and BA if possible, where

A =

1 2

3 4

B =

5 6

7 8

A is 2x2 and B is 2x2, so AB exists and BA exists.

AB =

1x5 + 2x7 1x6 + 2x8

3x5 + 4x7 3x6 + 4x8

=

19 22

43 50

(We have used the definition as follows:For example for the 1,2 entry of AB: Take row 1 of A and column 2 of B. Multiply corresponding elements and add: 1x6 + 2x8.)

BA =

5x1 + 6x3 5x2 + 6x4

7x1 + 8x3 7x2 + 8x4

=

23 34

31 46

EXAMPLE. Find AB and BA if possible, where

A =

1 2

3 4

5 6

B =

1 2 2 3

1 1 2 3

A is 3x2 and B is 2x4, so AB exists and is of size 3x4. Since B is 2x4 and A is 3x2, 4 does not equal 3, so BA does not exist.

AB =

1 + 2 2 + 2 2 + 4 3 + 6

3 + 4 6 + 4 6 + 8 9 + 12

5 + 6 10 + 6 10 + 12 15 + 18

=

3 4 6 9

7 10 14 21

11 16 22 33

PROPERTIES OF MATRIX MULTIPLICATION

VERY IMPORTANT: As the previous examples show AB and BA are not usually equal, and in fact one may exist whilst the other does not. Thus it is VERY important to be precise in the order in which we write the matrices that we are multiplying. Take particular notice of this in the properties below.

If the sizes of the matrices below are such that the following matrix multiplications are defined, then
(AB)C = A(BC)
(A + B)C = AC + BC
C(A + B) = CA + CB
0A = A0 = 0.

POWERS OF SQUARE MATRICES

We define
A² = AA, A³ = AAA, etc. Note that these are only defined if A is square.

The scene: Max is approaching the Identity Section of Matrix World.

So Max, do you think you're adept at multiplying matrices now?

Oh yes Trinity I did a thousand examples and got them all correct!

A thousand Max?

Well, would you believe a hundred? .... Well, how about ten?

OK Max, as long as you did a couple we'll have a look at identities now.

MATRIX IDENTITIES: EVEN MATRICES NEED AN IDENTITY!

A SQUARE matrix with all 1's on the main diagonal (from the upper left corner to the lower right corner) and 0's elsewhere is called an identity matrix. If it is an nxn matrix, we denote it I_n, but if the size is clear from the context we often don't bother to write the n. E.g.

I₂ =

1 0

0 1

I₃ =

1 0 0

0 1 0

0 0 1

I plays a similar role in matrix theory to the role 1 plays in real number multiplication. For matrices we have, if A is an nxn matrix, then

A I_n = I_nA = A.

Epilogue

Thats all pretty clear Trinity, what else do we have on matrices here?

Well Max, you could now go on and look at Matrix Inverse. Matrix inverses are very useful too.

I think I will! See you there folks! Bye for now.