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 m-by-n matrix or m×n matrix, while m and n are called its dimenstions. The above is a 4-by-3 matrix.

- Notaion of Matrix

Matrices are usually denoted using upper-case letters, while the corresponding lower-case letters, with two subscript indices, represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style, commonly boldface upright (non-italic), to further distinguish matrices from other variables.

The entry that lies in the i-th row and the j-th 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 i-th row of A, and a∗,j refers to the j-th column of A. The set of all m-by-n 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=A-B. 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 m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot-product 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.