Matrix (mathematics)

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In mathematics, a matrix (plural matrices) is a rectangular table of numbers. There are rules for adding, subtracting and "multiplying" matrices together. These rules sometimes lead to strange results. For example, if A and B are two matrices, it is not always true that A \cdot  B equals B \cdot A.

Matrices are so useful in science that they are taught in the first year of many university courses.

Contents

[edit] Definitions and notations

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 (or m×n matrix) and m and n are called its dimensions.

The places in the matrix where the numbers are, are called entry. The entry of a matrix A that lies in the row number i and column number j is called the i,j entry of A. This is written as A[i,j] or aij.

We write A:=(a_{ij})_{m \times n} to define an m × n matrix A with each entry in the matrix called aij for all 1 ≤ im and 1 ≤ jn.

[edit] Example

The matrix

\begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 7 \\ 4 & 9 & 2 \\ 6 & 1 & 5 \end{pmatrix}

is a 4×3 matrix. This matrix has m=4 rows, and n=3 columns.

The element A[2,3] or a23 is 7.

[edit] Operations

[edit] Addition

The sum of two matrices is the matrix, which (i,j)-th entry is equal to the sum of the (i,j)-th entries of two matrices:

\begin{pmatrix}     1 & 3 & 2 \\     1 & 0 & 0 \\     1 & 2 & 2   \end{pmatrix}   +   \begin{pmatrix}     0 & 0 & 5 \\     7 & 5 & 0 \\     2 & 1 & 1   \end{pmatrix}   =   \begin{pmatrix}     1+0 & 3+0 & 2+5 \\     1+7 & 0+5 & 0+0 \\     1+2 & 2+1 & 2+1   \end{pmatrix}   =   \begin{pmatrix}     1 & 3 & 7 \\     8 & 5 & 0 \\     3 & 3 & 3   \end{pmatrix}

You can see that the two matrices have the same dimensions. Here A + B = B + A is true.

[edit] Multiplication of two matrices

The multiplication of two matrices is more difficult:

\begin{pmatrix}     1 & 2 & 3 \\     4 & 5 & 6 \\   \end{pmatrix}   \cdot   \begin{pmatrix}     6 & -1 \\     3 & 2 \\     0 & -3   \end{pmatrix}   =   \begin{pmatrix}      (1 \cdot 6  +  2 \cdot 3  +  3 \cdot 0) &      (1 \cdot -1 +  2 \cdot 2 +  3 \cdot -3) \\      (4 \cdot 6  +  5 \cdot 3  +  6 \cdot 0) &      (4 \cdot -1 +  5 \cdot 2 +  6 \cdot -3) \\   \end{pmatrix}   =   \begin{pmatrix}     12 & -6 \\     39 & -12   \end{pmatrix}
  • two matrices can have different dimensions, but the number of columms of the first matrix is equal to the number of rows of the second matrix.
  • the product is a matrix with the same number of rows as the first matrix and the same number of columms as the second matrix.
  • the multiplication of matrices is not commutative, this means, in general is A \cdot  B  \neq B \cdot A
  • the multiplication of matrices is associative, this means (A \cdot B)\cdot C = A\cdot(B\cdot C)

[edit] Special matrices

There are some matrices that are special.

[edit] Square matrix

A square matrix has the same number of rows as columns, so m=n.

An example of a square matrix is

\begin{pmatrix}  5 & -2 & 4 \\  0 &  9 & 1 \\ -7 &  6 & 8 \\ \end{pmatrix}

This matrix has 3 rows and 3 columns: m=n=3.


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