In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices ...
, an augmented matrix is a
matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same
elementary row operations In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multi ...
on each of the given matrices.
Given the matrices and , where
the augmented matrix (''A'', ''B'') is written as
This is useful when solving
systems of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three equations in th ...
.
For a given number of unknowns, the number of solutions to a system of linear equations depends only on the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
of the matrix representing the system and the rank of the corresponding augmented matrix. Specifically, according to the
Rouché–Capelli theorem
In linear algebra, the Rouché–Capelli theorem determines the number of solutions for a system of linear equations, given the rank of its augmented matrix and coefficient matrix. The theorem is variously known as the:
* Rouché–Capelli theore ...
, any system of linear equations is
inconsistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
(has no solutions) if the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
of the augmented matrix is greater than the rank of the
coefficient matrix In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.
Coefficient matrix
In general, a system with ''m'' linear ...
; if, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has free parameters where is the difference between the number of variables and the rank; hence in such a case there are an infinitude of solutions.
An augmented matrix may also be used to find the inverse of a matrix by combining it with the
identity matrix.
To find the inverse of a matrix
Let be the square 2×2 matrix
To find the inverse of C we create (''C'', ''I'') where I is the 2×2
identity matrix. We then reduce the part of (''C'', ''I'') corresponding to ''C'' to the identity matrix using only
elementary row operations In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multi ...
on (''C'', ''I'').
the right part of which is the inverse of the original matrix.
Existence and number of solutions
Consider the system of equations
The coefficient matrix is
and the augmented matrix is
Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are an infinite number of solutions.
In contrast, consider the system
The coefficient matrix is
and the augmented matrix is
In this example the coefficient matrix has rank 2 while the augmented matrix has rank 3; so this system of equations has no solution. Indeed, an increase in the number of linearly independent rows has made the system of equations inconsistent.
Solution of a linear system
As used in linear algebra, an augmented matrix is used to represent the
coefficients
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
and the solution vector of each equation set.
For the set of equations
the coefficients and constant terms give the matrices
and hence give the augmented matrix
Note that the rank of the coefficient matrix, which is 3, equals the rank of the augmented matrix, so at least one solution exists; and since this rank equals the number of unknowns, there is exactly one solution.
To obtain the solution, row operations can be performed on the augmented matrix to obtain the identity matrix on the left side, yielding
so the solution of the system is .
References
* Marvin Marcus and Henryk Minc, ''A survey of matrix theory and matrix inequalities'',
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, book ...
, 1992, . Page 31.
{{Matrix classes
Matrices