# Using determinant to find constraints on equation

• I
• TheDemx27
In summary: I meant to say "In summary," not "In summary, Basically" Sorry about that.In summary, the conversation discusses using the determinant to find constraints for homogeneous equations. The specific determinant in question is sin^2(\theta )(1+cos^2(\theta )). The conversation also mentions using the determinant to solve for unknowns in a system of linear equations, with the example being solving for X and Y in a matrix equation.
TheDemx27
Gold Member

Basically I don't know how to get to the constraints from the system of equations. In class we used det to find constraints for homogenous equations, but we didn't go over this situation. Someone spell it out for me?

The determinant is $sin^2(\theta )(1+cos^2(\theta ))$.

mathman said:
The determinant is $sin^2(\theta )(1+cos^2(\theta ))$.
How did you get that?
##det(A)=cos^2(\theta )sin^2(\theta )+sin^4(\theta )## Which then simplifies into what they got: ## =\frac{1}{2}\ (1-cos(\theta ))=sin^2(\theta )##

My question is how they proceed after that anyways.

The solution for a particular unknown in a system of linear equations can be expressed as a ratio of two determinants. In your example, the determinant of the denominator is ##\sin^2(q)##. The numerator is the determinant of a matrix formed by using ##C_1, C_2## in place of some entries in the coefficient matrix, depending on which unknown you are solving for. See if the standard method of solving equations in that manner ( e.g. https://www.cliffsnotes.com/study-g...tions-using-determinants-with-three-variables ) derives the equations the article ends up with.##\begin{pmatrix} \cos(\theta) \sin(\theta) & \sin^2(\theta) \\ -\sin^2(\theta) & \cos(\theta)\sin(\theta) \end{pmatrix} \begin{pmatrix} X \\ Y \end{pmatrix} =\begin{pmatrix} C_1 \\ C_2 \end{pmatrix} ##

Solve for ##X,Y##.

TheDemx27
TheDemx27 said:
How did you get that?
##det(A)=cos^2(\theta )sin^2(\theta )+sin^4(\theta )## Which then simplifies into what they got: ## =\frac{1}{2}\ (1-cos(\theta ))=sin^2(\theta )##

My question is how they proceed after that anyways.
my mistake

## 1. How does using determinants help find constraints on equations?

The determinant of a matrix can be used to determine whether a system of equations has a unique solution, infinite solutions, or no solutions. This information can be used to identify constraints on the equations, such as when there are too few or too many variables in the system.

## 2. Can determinants be used to find constraints on any type of equation?

No, determinants can only be used to find constraints on systems of linear equations. Non-linear equations do not have determinants, and therefore, other methods must be used to find constraints on them.

## 3. How do you calculate the determinant of a matrix?

The determinant of a matrix is calculated by using a specific formula that involves the elements of the matrix. The formula varies depending on the size of the matrix, but it involves multiplying certain elements of the matrix and then adding or subtracting them to get the determinant value.

## 4. What does it mean when the determinant of a matrix is zero?

When the determinant of a matrix is zero, it means that the system of equations represented by the matrix has either infinite solutions or no solutions. This indicates that there are constraints on the equations, such as too many or too few variables.

## 5. Are there any limitations to using determinants to find constraints on equations?

Yes, there are limitations to using determinants. For example, determinants cannot be used to find constraints on non-linear equations. Additionally, if the matrix is very large, the calculation of the determinant can become very complex and time-consuming, making it impractical for certain applications.

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