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 ([latex]a_{i,\space j}[/latex]) of elements in A indexed by the Cartesian product of the sets of integers [1, m] and [1, n].
[latex]\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}[/latex]
- In the matrix above, the element [latex]a_{1, 2}[/latex] is read as "[latex]a[/latex] one-two".
- The first element of the ordered pair in the index indicates the row, whereas the second element indicates the column.
- The element [latex]a_{3, 2}[/latex] is located in the third row and in the second column.
Examples
- Consider the matrix : A = [latex]\begin{pmatrix} 3 & 6 & 7\\4 & 8 & 5\end{pmatrix}[/latex] Then : [latex]a_{1,2}[/latex] = 6 and [latex]a_{2,3}[/latex] = 5.
- Two matrices are equal if they have the same dimensions and if their corresponding elements are equal. Let A = [latex]\begin{pmatrix} –3 & 6 & 7\\4 & –8 & 5\end{pmatrix}[/latex] and B = [latex]\begin{pmatrix} x & 6 & 7\\4 & y & 5\end{pmatrix}[/latex]. Therefore : x = −3 and y = –8.
- The transpose of a matrix A of dimension m × n is the matrix B of dimension n × m such that [latex]b_{j,\space i}[/latex] = [latex]a_{i,\space j}[/latex].
- If A = [latex]\begin{pmatrix} –3 & 6 & 7\\4 & –8 & 5\end{pmatrix}[/latex], then B = [latex]\begin{pmatrix} –3 & 4\\6 & –8\\7 & 5\end{pmatrix}[/latex].