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In mathematics, a matrix
(plural matrices, or less commonly matrixes) is a rectangular array of numbers, as shown at the right. One use of
matrices is to keep track of the coefficients in a system of linear equations. Matrices can also represent linear
transformations, which are higher-dimensional analogs of linear functions of the form f(x) = cx, where c is a
constant. They can be added and subtracted entrywise, and multiplied according to a rule corresponding to
composition of linear transformations. These operations satisfy the usual identities, except that matrix
multiplication is not commutative: the identity AB=BA can fail. For a square matrix, the determinant and inverse
matrix (when it exists) govern the behavior of solutions to the corresponding system of linear equations, and
eigenvalues and eigenvectors provide insight into the geometry of the associated linear transformation.
Matrices find many applications. Physics makes use of them in various domains, for example in geometrical optics
and matrix mechanics. The latter also led to studying in more detail matrices with an infinite number of rows and
columns. Matrices encoding distances of knot points in a graph, such as cities connected by roads, are used in
graph theory, and computer graphics use matrices to encode projections of three-dimensional space onto a
two-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives of functions
or exponentials to matrices. The latter is a recurring need in solving ordinary differential equations. Serialism
and dodecaphonism are musical mouvements of the 20th century that utilize a square mathematical matrix (music) to
determine the pattern of music intervals. - Wikipedia
Diagonal Matrix with main diagonal elements = 1 |
All elements = 0 except for a single 1 |
Square Matrix with aij = aji |
| Same number of rows and columns |
Square Matrix with aij = 0 for all i¹ j |
All elements = 0 |
| Value of determinant = 0 |
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Square Matrix where aij is replaced with I, j cofactor multiplied by (-1)i·j. Adjoint A = adj A |
- Maximum number of linearly independent columns of the matrix A.
- Order of the longest nonsingular matrix contained in matrix A.
Rank of A = Rank of A' Rank of A = Rank of A'A
Rank of A = Rank of A A'
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Matrix where rows and columns are interchanged. Transpose A = A (A')' = A (kA)' = kA'
(A+B)' = A'+B' (AB)' = B'A' |
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