Day 1. About Matrices
Goal: For the students to recognize and understand the definition and terms of the matrix. 
 What is the Matrix?
A matrix is a rectangular arrangement of numbers. For example,
We can also use large parentheses as an alternative notaion instead of box brackets:
The horizontal and vertical lines in a matrix are called rows and columns. The numbers in the matrix are called its entries or its elememts.
To define a matrix's size, a matrix with m rows and n columns is called an mbyn matrix or m×n matrix, while m and n are called its dimenstions. The above is a 4by3 matrix.
 Notaion of Matrix
Matrices are usually denoted using uppercase letters, while the corresponding lowercase letters, with two subscript indices, represent the entries. In addition to using uppercase letters to symbolize matrices, many authors use a special typographical style, commonly boldface upright (nonitalic), to further distinguish matrices from other variables.
The entry that lies in the ith row and the jth column of a matrix is typically referred to as the i,j, (i,j), or (i,j)th entry of the matrix. For example, (2,3) entry of the above matrix A is 7. For example, the (i, j)th entry of a matrix A is most commonly written as aij. Alternative notations for that entry are A[ij] or Aij.
An asterisk is commonly used to refer to all of the rows or columns in a matrix. For example, ai,∗ refers to the ith row of A, and a∗,j refers to the jth column of A. The set of all mbyn matrices is denoted M(m, n). A common shorthand is A = [aij]i=1,...,m; j=1,...,n or more briefly A = [aij]m×n to define an m × n matrix A. Usually the entries aij are defined separately for all integers 1 ≤ i ≤ m and 1 ≤ j ≤ n.
 Opperation of Matrix
Operation  Definition  Example 

Addition  Denote the sum of two matrices A and B (of the same dimensions) by C=A+B. The sum is defined by adding entries with the same indices cij = aij + bij over all i and j.  
Subtract  Denote the subtaction of two matrices A and B (of the same dimensions) by C=AB. The subtract is defined by subtracting entries with the same indices cij = aij  bij over all i and j.  
multiply  <Scalar Multiplication> The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c. Denote: D=cA, dij = caij over all i and j. 

<Matrix Multiplication> Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an mbyn matrix and B is an nbyp matrix, then their matrix product AB is the mbyp matrix whose entries are given by dotproduct of the corresponding row of A and the corresponding column of B: where 1 ≤ i ≤ m and 1 ≤ j ≤ p. 
Writing out the product explicitly, where

There is a worksheet for matrices.