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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 matric ...
, an augmented matrix is a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
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-multipl ...
on each of the given matrices. Given the matrices and , where A = \begin 1 & 3 & 2 \\ 2 & 0 & 1 \\ 5 & 2 & 2 \end , \quad B = \begin 4 \\ 3 \\ 1 \end, the augmented matrix (''A'', ''B'') is written as (A, B) = \left begin 1 & 3 & 2 & 4 \\ 2 & 0 & 1 & 3 \\ 5 & 2 & 2 & 1 \end\right This is useful when solving systems of linear equations. For a given number of unknowns, the number of solutions to a system of linear equations depends only on the rank of the matrix representing the system and the rank of the corresponding augmented matrix. Specifically, according to the Rouché–Capelli theorem, any system of linear equations is inconsistent (has no solutions) if the rank of the augmented matrix is greater than the rank of the coefficient matrix; 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 In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
.


To find the inverse of a matrix

Let be the square 2×2 matrix C = \begin 1 & 3 \\ -5 & 0 \end. To find the inverse of C we create (''C'', ''I'') where I is the 2×2
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial ...
. 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-multipl ...
on (''C'', ''I''). (C, I) = \left begin 1 & 3 & 1 & 0\\ -5 & 0 & 0 & 1 \end\right (I, C^) = \left begin 1 & 0 & 0 & -\frac \\ 0 & 1 & \frac & \frac \end\right the right part of which is the inverse of the original matrix.


Existence and number of solutions

Consider the system of equations \begin x + y + 2z &= 2 \\ x + y + z &= 3 \\ 2x + 2y + 2z &= 6. \end The coefficient matrix is A = \begin 1 & 1 & 2 \\ 1 & 1 & 1 \\ 2 & 2 & 2 \\ \end, and the augmented matrix is (A, B) = \left begin 1 & 1 & 2 & 2\\ 1 & 1 & 1 & 3 \\ 2 & 2 & 2 & 6 \end\right 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 \begin x + y + 2z &= 3 \\ x + y + z &= 1 \\ 2x + 2y + 2z &= 5. \end The coefficient matrix is A = \begin 1 & 1 & 2 \\ 1 & 1 & 1 \\ 2 & 2 & 2 \\ \end, and the augmented matrix is (A, B) = \left begin 1 & 1 & 2 & 3 \\ 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 5 \end\right 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 \begin x + 2y + 3z &= 0 \\ 3x + 4y + 7z &= 2 \\ 6x + 5y + 9z &= 11 \end the coefficients and constant terms give the matrices A = \begin 1 & 2 & 3 \\ 3 & 4 & 7 \\ 6 & 5 & 9 \end , \quad B = \begin 0 \\ 2 \\ 11 \end, and hence give the augmented matrix (A, B) = \left begin 1 & 2 & 3 & 0 \\ 3 & 4 & 7 & 2 \\ 6 & 5 & 9 & 11 \end\right 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 \left begin 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -2 \\ \end\right 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, books ...
, 1992, . Page 31. {{Matrix classes Matrices