# Application of Cramer's Rule on linear system [closed]

I'm having some trouble with the following linear algebra question on Cramer's rule: An explanation for your choice of answer please.

Determine whether Cramer's Rule can be applied on the following system

$$\begin{cases}x_1\cos(u) - x_2\sin(u) = 1\\x_1\sin(u)+x_2\cos(u)=-3\end{cases}$$

Thanks in advance.

## closed as unclear what you're asking by user228113, SchrodingersCat, vrugtehagel, martini, drhabFeb 18 '16 at 16:01

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• Welcome to MSE! this site uses latex for math notations, please use it. Also, it is highly recommended to show some effort. What have you tried? – Galc127 Feb 18 '16 at 13:19
• Well, what is the condition in Cramer's rule (i.e. what needs to be satisfied for you to be allowed to apply it?) – Tobias Kildetoft Feb 18 '16 at 13:20
• It depends on which are the indeterminates and which are the parameters. – user228113 Feb 18 '16 at 13:25

## 1 Answer

To use Cramer's rule, the square coefficient matrix $A$ (for the system $AX=B$ with as many equations as unknowns) has to have a nonzero determinant.

What is $A$ for your system? Check its determinant.

Note: a nonzero determinant guarantees a unique solution (given by Cramer's rule).

• Why would the OP check that the determinant is $0$ when it is in fact not? – Tobias Kildetoft Feb 18 '16 at 13:34
• Thanks, my mistake! (edited) – StackTD Feb 18 '16 at 13:36
• Aaaah... Thanks... So finding the determinant of this coefficient matrix A by cofactor expansion along the first row, gives us (\cos\theta)(\cos\theta) + ((-\sin\theta)(-\sin\theta)) = \cos^2\theta + \sin^2\theta = 1 \$\therefore determinant of A is nonzero and Cramer's rule can be used. True. p.s. I don't know how to use latex. Will check it out soon. – Samiera Ebrahim Feb 18 '16 at 20:19
• That's correct! – StackTD Feb 18 '16 at 20:54