Discrete Math: Matrix Basics

Matrix Basics

• A matrix is a 2-dimensional array of numbers with rows and columns, represented by box brackets or parentheses.
• A matrix is an ordered set of numbers listed in rectangular form.
• The numbers in a matrix are called "entries" (or elements or components).

Matrix Dimensions and Size

• The size of a matrix with m rows and n columns is called an m x n matrix, while m and n are called its dimensions.
• A 3x3 matrix has 3 rows and 3 columns, with a total of 9 entries.
• A 3x4 matrix has 3 rows and 4 columns, with a total of 12 entries.

Types of Matrices

• A row vector is a matrix with one row.
• A column vector is a matrix with one column.
• A square matrix has the same number of rows and columns.
• A non-square matrix has a different number of rows and columns.
• A zero matrix (or null matrix) has all entries as zero, and can be of any dimension.
• A zero-one matrix has entries of only 0 and 1.
• A diagonal matrix is a square matrix with all entries as zero, except on the main diagonal.
• A tridiagonal matrix is a square matrix with all entries as zero, except on the main diagonal and the two adjacent diagonals.
• An upper triangular matrix has all entries underneath the main diagonal as zeroes.
• A lower triangular matrix has all entries above the main diagonal as zeroes.
• A scalar matrix has all terms on the main diagonal as equal.
• An identity matrix has all entries as zero, except those on the main diagonal as ones.

Matrix Operations

• Matrices can be added and subtracted if they have the same dimension (conformable).
• To add or subtract matrices, corresponding entries are added or subtracted.
• Matrices A and B are not conformable for addition or subtraction if they have different dimensions.

Properties of Matrix Addition

• Commutative law for addition: A+B = B+A
• Associative law for addition: (A+B)+C = A+(B+C)
• Additive identity: A+0 = A
• Additive inverse: A + (-A) = 0

Scalar Multiplication and Matrix Multiplication

• Scalar multiplication: multiplying a matrix by a scalar (number) multiplies each entry by that scalar.
• Matrix multiplication: the product of a row matrix and a column matrix.