Investigate for what values of λ and μ the system of linear equations So, the third row in the echelon form should be a zero row. In order that the system should have one parameter family of solutions, we must have ρ ( A) = ρ () = 2.
The matrix form of the system is AX = B, where A = If ρ ( A ) ≠ ρ (), then the system AX = B is inconsistent and has no solution.įind the condition on a, b and c so that the following system of linear equations has one parameter family of solutions: x + y + z = a, x + 2 y + 3z = b, 3x + 5 y + 7z = c.In particular, if there are 3 unknowns in a system of equations and ρ ( A ) = ρ () = 2, then the system has infinitely many solutions and these solutions form a one parameter family. In the same manner, if there are 3 unknowns in a system of equations and ρ ( A ) = ρ () = 1, then the system has infinitely many solutions and these solutions form a two parameter family. If there are n unknowns in the system AX = B, and ρ ( A ) = ρ () = n - k, k ≠ 0 then the system is consistent and has infinitely many solutions and these solutions form a k - parameter family.If there are n unknowns in the system of equations and ρ ( A ) = ρ () = n, then the system AX = B, is consistent and has a unique solution.By Rouché - Capelli theorem, we have the following rule: So the given system of equations is inconsistent and has no solution. If we write the equivalent system of equations using the echelon form, we get The matrix form of the system of equations is AX = B, whereĪpplying elementary row operations on the augmented matrix, we get Test the consistency of the following system of linear equations Here, the given system of equations is consistent and has infinitely many solutions which form a two parameter family of solutions. The above solution set is a two-parameter family of solutions. So, the solution is ( x = -9 + s - t, y = s, z = t ), where s and t are parameters.
Taking y = s, z = t arbitrarily, we get x - s + t = -9 or x = -9 + s - t.
The equivalent system has one non-trivial equation and three unknowns. The next example also confirms the supremacy of Gaussian elimination method over other methods. However, Gaussian elimination method is applicable and we are able to decide whether the system is consistent or not. In the above example, the square matrix A is singular and so matrix inversion method cannot be applied to solve the system of equations. Here, the given system is consistent and has infinitely many solutions which form a one parameter family of solutions. The above solution set is a one-parameter family of solutions. We fix z arbitrarily as a real number t , and we get y = 3 t - 2, x = -1- (3 t - 2) + 3 t = 1. So, the solution is ( x = 1, y = 3 t - 2, z = t ), where t is real. So, one of the unknowns should be fixed at our choice in order to get two equations for the other two unknowns. The equivalent system has two non-trivial equations and three unknowns. From the echelon form, we get the equivalent equations The matrix form of the system is AX = B, whereĪpplying elementary row operations on the augmented matrix, we get Here the given system is consistent and the solution is unique.Ĥ x − 2 y + 6 z = 8, x + y − 3 z = −1, 15 x − 3 y + 9 z = 21. So, the solution is (x = −1, y = 4, z = 4). By the method of back substitution, we get So ρ(A) = 3.įrom the echelon form, we write the equivalent system of equations There are three non-zero rows in the row-echelon form of. The matrix form of the system is AX = B, whereĪpplying Gaussian elimination method on, we get Test for consistency of the following system of linear equations and if possible solve: We apply the theorem in the following examples.
We state the following theorem without proof:Ī system of linear equations, written in the matrix form as AX = B, is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix that is, ρ ( A) = ρ (). In this section, we investigate it by using rank method. In second previous section, we have already defined consistency of a system of linear equation. Applications of Matrices: Consistency of System of Linear Equations by Rank Method
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