Results 1 to 3 of 3

Math Help - Isometry and determinant of matrix

  1. #1
    Newbie
    Joined
    Sep 2011
    Posts
    21

    Isometry and determinant of matrix

    Let e_1 = (1,0) and e_2 = (0,1). If T is an isometry of the plane fixing O, let T(e_1) = (a,b) and
    T(e_2) = (c,d) and let A be a 2x2 matrix such that the first row is (a c) and second row is (b d). Prove that det(A) = 1 or -1.

    Using the definition of isometry i obtained the following equations:
    a^2+b^2=1 ----------(1)
    c^2+d^2=1 ----------(2)
    ac+bd=0 -------------(3)

    However, i was unable to continue on to find det(A). Can this problem be solved just by using definition of isometry alone? Or do i need some results from linear algebra. (This problem was part of a group theory course)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Mar 2011
    Posts
    40

    Re: Isometry and determinant of matrix

    Quote Originally Posted by H12504106 View Post
    Let e_1 = (1,0) and e_2 = (0,1). If T is an isometry of the plane fixing O, let T(e_1) = (a,b) and
    T(e_2) = (c,d) and let A be a 2x2 matrix such that the first row is (a c) and second row is (b d). Prove that det(A) = 1 or -1.

    Using the definition of isometry i obtained the following equations:
    a^2+b^2=1 ----------(1)
    c^2+d^2=1 ----------(2)
    ac+bd=0 -------------(3)

    However, i was unable to continue on to find det(A). Can this problem be solved just by using definition of isometry alone? Or do i need some results from linear algebra. (This problem was part of a group theory course)

    \bullet \,\,a=0.

    \overset{(1)}{\Longrightarrow}b^2 = 1\Longrightarrow b=\pm 1\overset{(3)}{\Longrightarrow}d=0 \overset{(2)}{\Longrightarrow} c^2 = 1 \Longrightarrow c=\pm 1 \Longrightarrow det(A)=1 or det(A) = -1.

    \bullet \,\,a\neq 0.

    \overset{(3)}{\Longrightarrow} c=-\frac{bd}{a}\overset{(2)}{\Longrightarrow}\left(b^  2 + a^2 \right)d^2 = a^2\overset{(1)}{\Longrightarrow} d^2=a^2 \Longrightarrow d= \pm a.

    If d=a then by (3), b=-c so, det(A)=a^2 + b^2 \overset{(1)}{=}1 and if d=-a then b=c so, det(A) =-a^2 - b^2 \overset{(1)}{=}-1.
    Last edited by zoek; September 16th 2011 at 10:31 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2011
    Posts
    21

    Re: Isometry and determinant of matrix

    Thanks a lot!.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. what will be the determinant of the matrix A
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: November 25th 2011, 07:00 AM
  2. Determinant of a matrix
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: July 11th 2011, 01:57 AM
  3. [SOLVED] Derivative of a matrix inverse and matrix determinant
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 24th 2011, 09:18 AM
  4. [SOLVED] Determinant of 4 x 4 Matrix
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 12th 2010, 07:59 PM
  5. [SOLVED] 100 by 100 Matrix Determinant
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 19th 2008, 09:56 PM

Search Tags


/mathhelpforum @mathhelpforum