Similarly, a system of equations is said to be in " reduced row echelon form " or in " canonical form " if its augmented matrix is in reduced row echelon form.
12.
More specifically, according to the Rouch? Capelli theorem, any system of linear equations ( underdetermined or otherwise ) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix.
13.
Putting it another way, according to the Rouch? Capelli theorem, any system of equations ( overdetermined or otherwise ) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix.
14.
A linear system is consistent if and only if its coefficient matrix has the same rank as does its augmented matrix ( the coefficient matrix with an extra column added, that column being the column vector of constants ).
15.
The number of independent equations in a system equals the rank of the augmented matrix of the system & mdash; the system's coefficient matrix with one additional column appended, that column being the column vector of constants.
16.
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.
17.
But if the rank of " A " is only 1, then if the rank of the augmented matrix is 2 there is no solution but if its rank is 1 then all of the lines coincide with each other.
18.
In linear systems, indeterminacy occurs if and only if the number of independent equations ( the rank of the augmented matrix of the system ) is less than the number of unknowns and is the same as the rank of the coefficient matrix.
19.
Then if and only if the rank of the augmented matrix [ " A " | " b " ] is also 2, there exists a solution of the matrix equation and thus an intersection point of the " n " lines.
20.
As before there is a unique intersection point if and only if " A " has full column rank and the augmented matrix [ " A " | " b " ] does not, and the unique intersection if it exists is given by
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