- #1
johnstobbart
- 22
- 0
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?
Last edited: