## 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

*a*and called the

_{i,j}*i,j*entry or

*(i,j)*-th entry of

**A**. Alternative notations for that entry are

**A**[

*i,j*] or

**A**

_{i,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*c***A**is computed by multiplying every element of**A**by the scalar*c.*

Example:

Matrix Multiplication:

Matrix Multiplication:

Multiplication
of two matrices is well-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*×*n*matrix and**B**is an*m*×*n*matrix, then their**matrix product****AB**is the*m*×*p*matrix given by: