If you mean: 3x-4y = 19 and 3x-6y = 15
Then: x = 9 and y = 2
This is the case when there is only one set of values for each of the variables that satisfies the system of linear equations. It requires the matrix of coefficients. A to be invertible. If the system of equations is y = Ax then the unique solution is x = A-1y.
True
When the matrix of coefficients is singular.
Consider the linear system of equations AX = YwhereX is a n x 1 matrix of variables,Y is a n x 1 matrix of constants, andA is an n x n matrix of coefficients.Provided A is not a singular matrix, A has an inverse, A-1, an n x n matrix.Premultiplying by A-1 gives A-1AX = A-1Y or X = A-1Y, the solution to the linear system.
An independent system of linear equations is a set of vectors in Rm, where any other vector in Rm can be written as a linear combination of all of the vectors in the set. The vector equation and the matrix equation can only have the trivial solution (x=0).
An "inconsistent" set of equations. If they are all linear equations then the matrix of coefficients is singular.
This is the case when there is only one set of values for each of the variables that satisfies the system of linear equations. It requires the matrix of coefficients. A to be invertible. If the system of equations is y = Ax then the unique solution is x = A-1y.
Matrix inverses and determinants, square and nonsingular, the equations AX = I and XA = I have the same solution, X. This solution is called the inverse of A.
True
Create a matrix of the coefficients of each equation. The solutions to the equations should make up the rightmost column of the matrix. Then, row reduce the matrix until you are able to rewrite the equations and solve them. The matrix should be a 4x5 matrix (4 rows and 5 columns) for four equations with four variables. This is known as a system of equations.
Rank of a matrix is used to find consistency of linear system of equations.As we know most of the engineering problems land up with the problem of finding solution of a linear system of equations ,at that point rank of matrix is useful.
row reduce the matrix in question and see if it has any free variables. if it does then it has many solution's. If not then it only has one unique solution. which is of course the trivial solution (0)
constant matrix
When the matrix of coefficients is singular.
If one (or more) of the equations can be expressed as a linear combination of the others. This is equivalent to the statements the matrix of coefficients does not have an inverse (or is singular), or the determinant of the matrix of coefficients is zero.
A null matrix is a matrix with all its elements zero.EXAMPLES : (0 0) is a null row matrix.(0 0)(0 0) is a null square matrix.NOTE : Text handling limitations prevent the printing of large brackets to enclose the matrix array. Two pairs of smaller brackets have therefore been used.Answer 2:The above answer is a null matrix. However, the nullity of a matrix is the dimension of the kernel. Rank + Nullity = Dimension. So if you have a 4x4 matrix with rank of 2, the nullity must be 2. This nullity is the number of "free variables" you have. A 4x4 matrix is 4 simultaneous equations. If it is rank 2, you have only two independent equations and the other two are dependent. To solve a system of equations, you must have n independent equations for n variables. So the nullity tells you how short you are in terms of equations.
Consider the linear system of equations AX = YwhereX is a n x 1 matrix of variables,Y is a n x 1 matrix of constants, andA is an n x n matrix of coefficients.Provided A is not a singular matrix, A has an inverse, A-1, an n x n matrix.Premultiplying by A-1 gives A-1AX = A-1Y or X = A-1Y, the solution to the linear system.