I don't think the question actually makes sense.
See if you figure it out. In verbatim:
Suppose A is m by n and b is in R^n.
What has to be true about the two numbers'
rank[A b] and rankA for the equation Ax = b to be consistent?
Are the two "numbers" rank[A b], rankA?
Or is the question asking about something else?
See if you figure it out. In verbatim:
Suppose A is m by n and b is in R^n.
What has to be true about the two numbers'
rank[A b] and rankA for the equation Ax = b to be consistent?
Are the two "numbers" rank[A b], rankA?
Or is the question asking about something else?
-
If the first part is really verbatim, there is a typo in the question. If A is m x n, then Ax = b requires b to be in IR^m. The solution x, if one exists, would be in IRⁿ.
If this typo is fixed, then the question makes sense. The rank can be defined in several logically equivalent ways. A useful one from the perspective of this problem is that rank(M) is the number of nonzero rows in the reduced row echelon form of M. The number of nonzero rows is a positive integer. So they really are "numbers".
So rank(A) is the number of nonzero rows in rref(A). Similarly rank([A b]) is the number of nonzero rows in rref([A b]). Hopefully it is clear that [A b] is the augmented matrix obtained by adding on the column b to the matrix A.
Tacking on one additional column can not reduce the rank. So rank(A) ≤ rank([A b]). It turns out that if rank(A) < rank([A b]), then the system is not consistent. In terms of the system Ax = b, this would mean that Gaussian elimination would lead to an equation of the form
0x1 + 0x2 + ... + 0xn = 1 --- i.e. 0 = 1
which clearly has no solution.
So the answer is that the system is consistent if and only if rank(A) = rank([A b]).
If this typo is fixed, then the question makes sense. The rank can be defined in several logically equivalent ways. A useful one from the perspective of this problem is that rank(M) is the number of nonzero rows in the reduced row echelon form of M. The number of nonzero rows is a positive integer. So they really are "numbers".
So rank(A) is the number of nonzero rows in rref(A). Similarly rank([A b]) is the number of nonzero rows in rref([A b]). Hopefully it is clear that [A b] is the augmented matrix obtained by adding on the column b to the matrix A.
Tacking on one additional column can not reduce the rank. So rank(A) ≤ rank([A b]). It turns out that if rank(A) < rank([A b]), then the system is not consistent. In terms of the system Ax = b, this would mean that Gaussian elimination would lead to an equation of the form
0x1 + 0x2 + ... + 0xn = 1 --- i.e. 0 = 1
which clearly has no solution.
So the answer is that the system is consistent if and only if rank(A) = rank([A b]).
-
Bases don't have ranks. Matrices have ranks. If b is in IRⁿ, then it is true that m = n. But it does not follow that rank(A) = n. All you can say is rank(A) ≤ n. Moreover, it is possible that rank([A b]) > rank(A). In this case the system is not consistent. A and [A b] aren't numbers.
Report Abuse