You are already familiar with vectors, which are an array of numbers. Matrices are a rectangular array of numbers. The following is an example of a matrix:

This matrix has 2 rows and 3 columns. Therefore it is called a 2×3 matrix. A vector can be considered a special case of a matrix that has only one column. In general a matrix looks like this:

where is the entry in row and column . This matrix has rows and columns.

In the following, we describe different special kinds of matrices and other terminology.

Square matrix
A matrix that has the same number of rows as columns () is called a square matrix. For example, the following is a square matrix:

Diagonal
The entries (, , …, ) are said to be on the diagonal. All other entries are off-diagonal. The following is an example of a matrix that non-zero entries on the diagonal and all off-diagonal entries are zero:

Such a matrix that contains many zeros can be abbreviated as:

Identity matrix
A matrix where all off-diagonal entries are zero, and all diagonal entries are one is called an identity matrix:

Triangular matrix
A matrix where all entries below or above the diagonal are zero is said to be triangular. If all entries below are zero, it is called upper triangular. If all entries above are zero, it is called lower triangular. This is an example of an upper triangular matrix:

Operations with matrices

For more information on how to calculate with matrices, please refer to the following articles:

• matrix transpose
• matrix addition
• matrix multiplication with a scalar
• matrix multiplication
• matrix inverse