A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. The numbers, symbols, or expressions in the matrix are called its entries or its elements. Matrices are subject to standard operations such as addition and multiplication.
There are many different types of matrices, each with its own unique properties. Some of the most common types of matrices include:
- Row matrix: A row matrix has only one row but any number of columns. For example, the matrix
[1, 2, 3]is a row matrix. - Column matrix: A column matrix has only one column but any number of rows. For example, the matrix
[1; 2; 3]is a column matrix. - Square matrix: A square matrix has the same number of rows and columns. For example, the matrix
[1, 2; 3, 4]is a square matrix. - Diagonal matrix: A diagonal matrix has all zeroes off the main diagonal and non-zeroes on the main diagonal. For example, the matrix
[1, 0; 0, 1]is a diagonal matrix. - Upper triangular matrix: An upper triangular matrix has all zeroes below the main diagonal. For example, the matrix
[1, 2, 3; 0, 4, 5; 0, 0, 6]is an upper triangular matrix. - Lower triangular matrix: A lower triangular matrix has all zeroes above the main diagonal. For example, the matrix
[1, 0, 0; 2, 4, 0; 3, 5, 6]is a lower triangular matrix. - Symmetric matrix: A symmetric matrix is equal to its transpose. For example, the matrix
[1, 2; 2, 4]is a symmetric matrix. - Skew-symmetric matrix: A skew-symmetric matrix is equal to the negative of its transpose. For example, the matrix
[1, -2; 2, 1]is a skew-symmetric matrix. - Identity matrix: An identity matrix is a square matrix with ones on the main diagonal and zeroes elsewhere. For example, the matrix
[1, 0, 0; 0, 1, 0; 0, 0, 1]is an identity matrix.
These are just a few of the many different types of matrices. Matrices are a powerful tool for representing and solving problems that involve multiple variables. They are used in a wide variety of fields, including mathematics, physics, chemistry, engineering, computer science, and economics.