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

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