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Hello everyone, I have a linear algebra question regarding Cramer's rule.

Using Cramer's rule, solve for x' and y' in terms of x and y.

[tex]

\begin{cases}

x = x' cos \theta - y' sin \theta\\

y = x' sin \theta + y'cos \theta

\end{cases}

[/tex]

2. Homework Equations

##sin^2 \theta + cos^2 \theta = 1 ##

I need a matrix to start off, so I form a matrix based on the right-hand side of x and y. I'm assuming that x' and y' are just alternative ways of writing ##x_1## and ##x_2##.

[tex]

Let A =

\begin{bmatrix}

cos \theta & -sin \theta\\

sin \theta & cos\theta

\end{bmatrix}

[/tex]

I form two more matrices, ##A_1## and ##A_2##.

[tex]

Let A_1 =

\begin{bmatrix}

x & -sin \theta\\

y & cos\theta

\end{bmatrix}

[/tex]

[tex]

Let A_2 =

\begin{bmatrix}

cos \theta & x\\

sin \theta & y

\end{bmatrix}

[/tex]

I then find ##det(A)##. I get ##cos^2 \theta + sin^2 \theta ## which is ##1##.

##det(A_1) = x cos \theta + y sin \theta##

##det(A_2) = y cos \theta - x sin \theta##

Lastly, I need to find the value of ##x'## and ##y'##, using Cramer's Rule.

[tex]

x' = \frac{det(A_1)}{det A} = \frac{x cos \theta + y sin \theta}{1}\\

y' = \frac{det(A_2)}{det A} = \frac{y cos \theta - x sin \theta}{1}

[/tex]

Can anyone tell me if I'm on the right track for this problem?

## Homework Statement

Using Cramer's rule, solve for x' and y' in terms of x and y.

[tex]

\begin{cases}

x = x' cos \theta - y' sin \theta\\

y = x' sin \theta + y'cos \theta

\end{cases}

[/tex]

2. Homework Equations

##sin^2 \theta + cos^2 \theta = 1 ##

## The Attempt at a Solution

I need a matrix to start off, so I form a matrix based on the right-hand side of x and y. I'm assuming that x' and y' are just alternative ways of writing ##x_1## and ##x_2##.

[tex]

Let A =

\begin{bmatrix}

cos \theta & -sin \theta\\

sin \theta & cos\theta

\end{bmatrix}

[/tex]

I form two more matrices, ##A_1## and ##A_2##.

[tex]

Let A_1 =

\begin{bmatrix}

x & -sin \theta\\

y & cos\theta

\end{bmatrix}

[/tex]

[tex]

Let A_2 =

\begin{bmatrix}

cos \theta & x\\

sin \theta & y

\end{bmatrix}

[/tex]

I then find ##det(A)##. I get ##cos^2 \theta + sin^2 \theta ## which is ##1##.

##det(A_1) = x cos \theta + y sin \theta##

##det(A_2) = y cos \theta - x sin \theta##

Lastly, I need to find the value of ##x'## and ##y'##, using Cramer's Rule.

[tex]

x' = \frac{det(A_1)}{det A} = \frac{x cos \theta + y sin \theta}{1}\\

y' = \frac{det(A_2)}{det A} = \frac{y cos \theta - x sin \theta}{1}

[/tex]

Can anyone tell me if I'm on the right track for this problem?

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