Find the orbit and stabilizer of the 2 X 2 matrix M under the action of multiplication of M by the matrices in GL_2(R), where the top row of M is (1 0) and the bottom row is (0 2). [That is, m_11 = 1, m_12 = 0, m_21 = 0, and m_22 = 2.]
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Recall that GL_2(R) is the set of 2 X 2 invertible matrices whose elements are real numbers (the "R" in "GL_2(R)" denotes the set of real numbers).
The orbit of M under the action of GL_2(R) is the set of all 2 X 2 matrices which is obtained by multiplying the matrices in GL_2(R) by M.
Let Y be a matrix in GL_2(R), and denote the top and bottom rows of Y by (a b) and (c d), respectively, where a, b, c, and d are real numbers. [That is, y_11 = a, y_12 = b, y_21 = c, and y_22 = d.] Since Y must be invertible, the determinant of Y, which is equal to (a*d) - (b*c), must be non-zero; hence (a*d) must be unequal ...
The definitions of orbit of matrix and stabilizer of matrix (under the action of multiplication by matrices of the specified type) are reviewed, and a detailed solution of the problem is provided.