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 equals
.
Matrices are so useful in science that they are taught in the first year of many university courses.
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[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 to define an m × n matrix A with each entry in the matrix called aij for all 1 ≤ i ≤ m and 1 ≤ j ≤ n.
[edit] Example
The matrix
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:
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:
- 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
- the multiplication of matrices is associative, this means
[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
This matrix has 3 rows and 3 columns: m=n=3.
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