# Matrix Definition. Main information

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## Matrix definition

Definition.

**Matrix**with size n×m is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns, which consisting of n rows and m columns.

The number of rows and columns are defined the matrix size.

## Matrix notation

Matrices are commonly written in *box brackets* or *large parentheses*:

A = | 4 | 1 | -7 | ||

-1 | 0 | 2 |

The matrix usually denoted by capital letters of the Latin alphabet. The matrix comprising n rows and m columns, called the matrix of size n×m. Also, the size of the matrix is written following way: **A _{n×m}**.

## Matrix entries

Entries of the matrix

**A**are denoted a_{ij}, where i is row number, in which is an element, j is the column number.Example.

Entries of matrix A_{4×4}:

A = | 4 | 1 | -7 | 2 | ||

-1 | 0 | 2 | 44 | |||

4 | 6 | 7 | 9 | |||

11 | 3 | 1 | 5 |

a_{1}_{1} = 4

Definition.

Matrix row is **zero**if all of its elements equal to zero.

Definition.

If at least one of the elements of the matrix row is not zero, the row is called a **non-zero**.

Example.

Zero and non-zero matrix row:4 | 1 | -7 | < non-zero matrix row | ||

0 | 0 | 0 | |||

0 | 1 | 0 |

Definition.

Matrix column is **zero**if all of its elements equal to zero.

Definition.

If at least one of the elements of the matrix column is not zero, the column is called a **non-zero**.

Example.

Zero and non-zero matrix column:0 | 1 | -7 | ||

0 | 0 | 2 | ||

^ | ^ | ^ |

non-non-zero matrix column

## Matrix diagonal

Definition.

**Main diagonal of matrix**is the collection of entries a

_{ij}where i = j.

Definition.

**Antidiagonal of matrix**with size n×m is the collection of entries a

_{ij}where i + j = n + 1.

Example.

Main diagonal and antidiagonal of matrix:0 | 1 | -7 | - main diagonalantidiagonal of matrix | ||

0 | 0 | 2 |

0 | 1 | -7 | - main diagonalantidiagonal of matrix | ||

0 | 0 | 2 | |||

8 | 2 | 9 |

Definition.

The **trace**of square matrix A is defined to be the sum of the elements on the main diagonal.

Definition.

The **trace of matrix**is denoted as

tr(**A**) = a_{11} + a_{22} + ... + a_{nn}.

MatrixMatrix Definition. Main informationSystem of linear equations - matrix formTypes of matricesMatrix scalar multiplicationAddition and subtraction of matricesMatrix multiplicationTranspose matrixElementary matrix operationsDeterminant of a matrixMinors and cofactors of a matrixInverse matrixLinearly dependent and independent rowsRank of a matrix

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