ZedCar said:Homework Statement
Does the following set of linear equations have a unique solution?
(excluding any trivial solutions when x=y=z=0)
Do not attempt to formally solve the equations.
x+2y-4z=8
4x-6y+12z=19
-6x+3y-6z=-20
Homework Equations
The Attempt at a Solution
My immediate thought would be to solve the equations using Cramers Rule. However, it states not to formally solve the equations. So I assume I'm not allowed to do this.
So what does the question mean me to do?
Yes.ZedCar said:By ^T do you mean transpose?
Just wanted to check before attempting to try and solve this.
A better way is to say that it has no solution. A system of equations can haveZedCar said:If I take the matrix;
1, 2, -4
4, -6, 12
-6, 3, -6
The determinant = 0
Therefore it is not invertible.
Therefore this implies it has no unique solution.
Is this correct?
For this particular problem, there is no solution. I am distinguishing between "no unique solution" and "no solution" as the former term might be interpreted by some to mean that there are multiple solutions (i.e., not a unique solution). That was the distinction I was trying to make.HallsofIvy said:But isn't "no unique solution" what he wants to say? The question, after all, was "Does the following set of linear equations have a unique solution?" The fact that the determinant of this 3 by 3 set of equations is 0 means it maps R3 into a proper subspace of R3. If the right hand side doesn't happen to be in that subspace, there is no solution but if it does, an entire subspace will be mapped to it.
Cramer's Rule is a method for solving a system of linear equations by using determinants. It was named after Swiss mathematician Gabriel Cramer.
Cramer's Rule is typically used when solving systems of linear equations with two or three variables. It can also be used to find the solution of a single variable in a system of equations.
To use Cramer's Rule, the coefficients of the variables in each equation must be arranged in a matrix. The determinant of this matrix is then calculated, along with the determinants of matrices formed by replacing the coefficients of one variable with the constants from the equations. The solution is then found by dividing each of these determinants by the main determinant.
The main advantage of Cramer's Rule is that it provides a method for solving systems of linear equations without needing to manipulate the equations themselves. However, it is not efficient for large systems and can be prone to error due to the calculations involved in finding determinants.
No, Cramer's Rule can only be used for systems of linear equations. It cannot be used for non-linear equations, as these do not have constant coefficients and therefore cannot be represented in a matrix form.