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In the preceding tutorial we learned how to add matrices and multiply matrices by scalars (real numbers). For example,

Corresponding entries added  

Each entry multiplied by 4 
Q What about matrix multiplication? Do we multiply matrices as follows?

? 
Q OK. What is the "correct way"?
A Since the correct way may seem a little strange at first, let us do it stepbystep.
Q How can we apply this strange method of multiplying rows by columns?
A Here is an application: Suppose you sell 3 Tshirts at $10 each, 4 hats at $15 each, and 1 pair of shorts at $20. Then your total revenue is


=  [103 + 154 + 120]  =  [110]  
Price  Quantity  =  Revenue 
Q Now give me an application for multiplying matrices with more than one row and/or column.
A Here is an application that extends
the application above: Suppose you make the following sales:


=  [110 130]  
Price  Quantity  =  Revenue 
There are tons of other applications in Exercise Set 3.2 of Finite Mathematics and Finite Mathematics & Applied Calculus Also see theonline review exercises.
Identity Matrices
An identity matrix is a square matrix, called I, that has ones down the leading diagonal (see the examples below) and zeros everywhere else.
Here are some identity matrices.
I = [ 1 ]  11 identity matrix  

22 identity matrix  

33 identity matrix  

44 identity matrix 
Note The leading diagonal of a square matrix is the diagonal from the top left to the bottom right. It consists of the entries a_{11}, a_{22}, a_{33}, ..., a_{nn}.
Q Why are these matrices called "identity matrices"?
A Let us mutliply an identity matrix by some other matrix and see what we get:
In other words, is seems that that, if A is any square matrix, then
In other words, multiplying a matrix by I leaves it unchanged. This is analogous to multiplying a number, a, by 1. You should now try multiplying a 2 2 matrix by the 2 2 identity matrix. You will once again end up with the matrix you started with.
Q What happens if you multiply them the other way around? In other words, what is A^{.}I? (Remember that matrix multiplcation is not commutative)
A Try it out, and you will find once again that the result is A. In other words,
for every square matrix A.
Here is a list of rules that summarize the multiplicative properties of matrices.
Now try some of the exercises in Section 3.1 of Finite Mathematics or Finite Mathematics and Applied Calculus