Skip to content
Resources

    Matrices


    A matrix is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied.  

    The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with m rows and n columns is called an m-by-n matrix (written m × n) and m and n are called its dimensions.

    The dimensions of a matrix are always given with the number of rows first, then the number of columns. It is commonly said that an m-by-n matrix has an order of m × n ("order" meaning size). Two matrices of the same order whose corresponding entries are equivalent are considered equal.



    Almost always capital letters denote matrices with the corresponding lower-case letters with two indices representing the entries. For example, the entry of a matrix A that lies in the i-th row and the j-th column is written as ai,j and called the i,j entry or (i,j)-th entry of A. Alternative notations for that entry are A[i,j] or Ai,j. The row is always noted first, then the column.

    A matrix where one of the dimensions equals one is often called a vector, and interpreted as an element of real coordinate space. An m × 1 matrix (one column and m rows) is called a column vector and a 1 × n matrix (one row and n columns) is called a row vector.

    Matrix Addition:

    Two or more matrices of identical dimensions m and n can be added. Given m-by-n matrices A and B, their sum A+B is the m × n matrix computed by adding corresponding elements.

    Example:



    Matrix Subtration:

    Similar to matrix addition as described above, two or more matrices of identical dimensions m and n can be subtracted. Given m × n matrices A and B, their sum A-B is the m × n matrix computed by adding corresponding elements.


    Scalar Mulitplication:

    Given a matrix A and a number c, the scalar multiplication cA is computed by multiplying every element of A by the scalar c.

    Example:




    Matrix Multiplication: