Matrix formed by appending columns of two other matrices
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In linear algebra, an augmented matrix is a matrix obtained by appending a -dimensional row vector , on the right, as a further column to a -dimensional matrix . This is usually done for the purpose of performing the same elementary row operations on the augmented matrix as is done on the original one when solving a system of linear equations by Gaussian elimination.
For example, given the matrices and column vector , where
the augmented matrix is
For a given number of unknowns, the number of solutions to a system of linear equations depends only on the rank of the matrix of coefficients representing the system and the rank of the corresponding augmented matrix where the components of consist of the right hand sides of the successive linear equations. According to the Rouché–Capelli theorem, any system of linear equations
where is the -component column vector whose entries are the unknowns of the system 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. In such a case there as an affine space of solutions of dimension equal to this difference.
The inverse of a nonsingular square matrix of dimension may be found by
appending the identity matrix to the right of to form the dimensional augmented matrix . Applying elementary row operations to transform the left-hand block to the identity matrix , the right-hand block is then the inverse matrix
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otherwise) is inconsistent if the rank of the augmentedmatrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these...
consistent if and only if the coefficient matrix has the same rank as its augmentedmatrix (the coefficient matrix with an extra column added, that column...
transformations such that the 2×2 matrix of the A i , j {\displaystyle A_{i,j}} is orthogonal. The augmentedmatrix that represents the composition of...
system of linear equations is said to be in row echelon form if its augmentedmatrix is in row echelon form. Similarly, a system of linear equations is...
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otherwise) is inconsistent if the rank of the augmentedmatrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these...
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consistent if and only if its coefficient matrix has the same rank as does its augmentedmatrix (the coefficient matrix with an extra column added, that column...
F}\|[E\;F]\|_{F},\qquad (X+E)B=Y+F} where [ E F ] {\displaystyle [E\;F]} is the augmentedmatrix with E and F side by side and ‖ ⋅ ‖ F {\displaystyle \|\cdot \|_{F}}...
Numbers t1, t2, ..., tk such that v = t1b1 + ··· + tkbk Create an augmentedmatrix A whose columns are b1,...,bk , with the last column being v. Use elementary...
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![{\displaystyle (A|B)=\left[{\begin{array}{ccc|c}1&3&2&4\\2&0&1&3\\5&2&2&1\end{array}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f14272674b4b17491481e7ae9bcfcbc168ccb58)
