Ch1_5.doc
Rank of a Matrix
The rank of a matrix is the order of the largest nonzero determinant that can be obtained from the elements of the matrix (for both rectangular and square matrices).
The rank of a matrix is the maximum number of linearly independent rows or columns.
The rank of a matrix remains unchanged if (1) a multiple of one row (column) is added to another row (column); (2) rows (or columns) are interchanged.
A matrix A can be brought into a form in which only entries with the same row and column index can be different form zero (similar to Gaussian elemination). The rank of A is the number of such non-zero entries.
Example 1.5-5 ______
Find the rank of the matrix A where
A =
Solution
13123(row 1*)
(1)(row 1*) + (row 2)01211(row 2*)
(2)(row 1*) + (row 3)03633(row 3*)
(3)(row 1*) + (row 4)01211(row 4*)
13123(row 1)
01211(row 2)
(3)(row 2) + (row 3*)00000(row 3*)
(1)(row 2) + (row 4*)00000(row 4*)
The rank of matrix A is 2.
-A square matrix of order n has a rank less than n if the matrix is singular.
-The value of a determinant is zero if one row is a linear combination of other rows. The rows are linearly dependent.
Conditions for the Solution of Linear Equations
Ax = b
where
A = coefficient matrix =
b = right hand side vector =
The augmented matrix is defined as
[A| b] =
The form of the solution to Ax = b can be determined by investigating the rank of A and [A| b]. For n equation and n unknowns
If rank(A) = n then rank([A| b]) = n
rank(A) = rank([A| b])
If rank([A| b]) > rank(A) then rank([A| b]) = rank(A) + 1
A set of simultaneous equations is called consistent if there is a solution even a trivial solution. A set of simultaneous equations is called inconsistent if there is no solution. Table 1.5-1 summarizes the conditions for solution of n simultaneous equations.
Table 1.5-1 Conditions for Solution of n Simultaneous Equations
Type of Equations / Conditions of Rank / Nature of SolutionConsistent equations / rank(A) = rank([A| b]) = n / Unique solution
rank(A) = rank([A| b]) = r n / Infinite number of solutions with n-r arbitrary constants
Inconsistent equations / rank(A) < rank([A| b]) / No solution
Homogeneous equations
(special case of consistent equations ) / rank(A) = rank([A| b]) = n / Unique solution (the trivial solution)
rank(A) = rank([A| b]) = r n / Infinite number of solutions with n-r arbitrary constants
The five cases for the solution of n simultaneous equations will be presented with numerical examples using two equations.
- Unique solution for non-homogeneous equations: rank(A) = rank([A| b]) = n
x1 + 4 x2 = 8
x1 + x2 = 7
A = ,[A| b] =
det (A) = 5 0 rank(A) = rank([A| b]) = 2
x =
Figure 1.5-1 Unique solution of two simultaneous equations
2.Infinite number of solutions with n-r arbitrary constants: rank(A) = rank([A| b]) = r n
3x1 + 2 x2 = 12
6x1 4 x2 = 24
rank(A) = rank([A| b]) = 1 < 2 infinite solution with 2 1 = 1 arbitrary constant.
x = = = x2+ = x2+
where x2 = arbitrary constant
Figure 1.5-2 Infinite solution with one arbitrary constant
3.Inconsistent equations (No solution): rank(A) < rank([A| b])
x1 3 x2 = 3
x1 3 x2 = 18
A = det(A) = 0 rank(A) = 1
[A| b] = , = 15 rank([A| b]) = 2 no solution
Figure 1.5-3 Two parallel equations with no solution (intersection)
4. Homogeneous equations: rank(A) = rank([A| b]) = n. Unique solution (the trivial solution)
2x1 5 x2 = 0
x1 x2 = 0
rank(A) = rank([A| b]) = 2 x = = 0
Figure 1.5-4 Two homogeneous equations with trivial solution
- Homogeneous equations: rank(A) = rank([A| b]) = r n. Infinite number of solutions with n-r arbitrary constants.
x1 2 x2 = 0
3x1 + 6 x2 = 0
rank(A) = rank([A| b]) = 1 infinite solution with 2 1 = 1 arbitrary constant.
x = = = x2
Figure 1.5-5 Two homogeneous equations with infinite solution
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