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 × n 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 = \(\begin{pmatrix} –3 & 4\\6 & –8\\7 & 5\end{pmatrix}\).