# Matrix

## Matrix

An ordered array of numbers.

A matrix A of dimension m × n is a table that contains m rows and n columns in which numbers are in an m·n arrangement.

Let A be a set of numbers and (m, n) be an ordered pair of positive integers. The coefficient matrix in A, of dimension m × n, that is, of m rows and n columns, is a family ($$a_{i,\space j}$$) of elements in A indexed by the Cartesian product of the sets of integers [1, m] and [1, n].

$$\begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & … & a_{1,n}\\a_{2,1} & a_{2,2} & a_{2,3} & … & a_{2,n}\\a_{3,1} & a_{3,2} & a_{3,3} & … & a_{3,n}\\ … & … & … & … & …\\a_{m,1} & a_{m,2} & a_{m,3} & … & a_{m,n}\end{pmatrix}$$

• In the matrix above, the element $$a_{1, 2}$$ is read as “$$a$$ one-two”.
• The first element of the ordered pair in the index indicates the row, whereas the second element indicates the column.
• The element $$a_{3, 2}$$ is located in the third row and in the second column.

### Examples

• Consider the matrix : A = $$\begin{pmatrix} 3 & 6 & 7\\4 & 8 & 5\end{pmatrix}$$
Then : $$a_{1,2}$$ = 6 and $$a_{2,3}$$ = 5.
• Two matrices are equal if they have the same dimensions and if their corresponding elements are equal.
Let A = $$\begin{pmatrix} –3 & 6 & 7\\4 & –8 & 5\end{pmatrix}$$ and B = $$\begin{pmatrix} x & 6 & 7\\4 & y & 5\end{pmatrix}$$. Therefore : x = −3 and y = –8.
• The transpose of a matrix A of dimension m × is the matrix B of dimension n × m such that $$b_{j,\space i}$$ = $$a_{i,\space j}$$.
• If A = $$\begin{pmatrix} –3 & 6 & 7\\4 & –8 & 5\end{pmatrix}$$, then B = $$\begin{pmatrix} –3 & 4\\6 & –8\\7 & 5\end{pmatrix}$$.