Matrix (mathematics)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a matrix (pl.: matrices) is a <a href="/facts/Rectangle/mdLfIGKA">rectangular</a> array of <a href="/facts/Number/auQSAE2m">numbers</a> or other <a href="/facts/Mathematical_objects/Juhyow9M">mathematical objects</a> with elements or entries arranged in rows and columns, usually satisfying certain properties of <a href="/facts/Matrix_addition/6zzLWyL0">addition</a> and <a href="/facts/Matrix_multiplication/lYDB6Ro2">multiplication</a>. 
For example,

[
            
              
                
                  1
                
                
                  9
                
                
                  −
                  13
                
              
              
                
                  20
                
                
                  5
                
                
                  −
                  6
                
              
            
            ]
          
        
      
    
    {\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}}

denotes a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a "⁠
 
 
 
 2
 ×
 3
 
 
 {\displaystyle 2\times 3}
 
⁠ matrix", or a matrix of dimension ⁠
 
 
 
 2
 ×
 3
 
 
 {\displaystyle 2\times 3}
 
⁠.
In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, matrices are used as <a href="/facts/Linear_map/Q1PBKNVV">linear maps</a>. In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, matrices are used for <a href="/facts/Geometric_transformation/qRjrasJf">geometric transformations</a> (for example <a href="/facts/Rotation_(mathematics)/6gtkkcy0">rotations</a>) and <a href="/facts/Coordinate_change/Mtd0q4SP">coordinate changes</a>. In <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly, or through their use in geometry and numerical analysis.
<a href="/facts/Square_matrices/5TP9mNNn">Square matrices</a>, matrices with the same number of rows and columns, play a major role in matrix theory. The <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of a square matrix is a number associated with the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a> if and only if it has a nonzero determinant and the <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> of a square matrix are the roots of a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> determinant.
Matrix theory is the <a href="/facts/Branch_of_mathematics/5db06g8z">branch of mathematics</a> that focuses on the study of matrices. It was initially a sub-branch of <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, but soon grew to include subjects related to <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>, <a href="/facts/Algebra/nMdqczjo">algebra</a>, <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>.

Matrix (mathematics) open-in-new

Matrix (mathematics)