Question

2.3 Bonus: The determinant as a homomorphism The set GLn(R) of invertible matrices with real entries forms a group. The determinant of a matrix can be thought of as a map det : GLn(R) ? R* where R* is the group R ? 0 under multiplication. Since det AB = det A det B, det is a homomorphism. a) Show that the set L of matrices with the form M = (a ?b) (b a) where det M ? 0 is a subgroup of GL2(R). b) Give a geometric description of the quotient group L/ker(det). (Of course, we're thinking of det as map from L to R*.)

          2.3 Bonus: The determinant as a homomorphism
The set GLn(R) of invertible matrices with real entries forms a group. The determinant of a matrix can be thought of as a map
det : GLn(R) ? R*
where R* is the group R ? 0 under multiplication. Since det AB = det A det B, det is a homomorphism.
a) Show that the set L of matrices with the form
M = (a ?b)
    (b  a) where det M ? 0
is a subgroup of GL2(R).
b) Give a geometric description of the quotient group L/ker(det). (Of course, we're thinking of det as map from L to R*.)

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2.3 Bonus: The determinant as a homomorphism
The set GLn(R) of invertible matrices with real entries forms a group. The determinant of a matrix can be thought of as a map
det : GLn(R) ? R*
where R* is the group R ? 0 under multiplication. Since det AB = det A det B, det is a homomorphism.
a) Show that the set L of matrices with the form
M = (a ?b)
(b a) where det M ? 0
is a subgroup of GL2(R).
b) Give a geometric description of the quotient group L/ker(det). (Of course, we're thinking of det as map from L to R*.)

Added by Peter K.

Mathematical Methods for Physics and Engineering

K. F. Riley, M. P. Hobson, S. J. Bence 3rd Edition

Chapter 28

Instant Answer

Solved by Expert Sri K

10/21/2023

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a) To show that L is a subgroup of GL2(R), we need to show that it satisfies the three conditions of a subgroup: Show more…

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2.3 Bonus: The determinant as a homomorphism The set GLn(R) of invertible matrices with real entries forms a group. The determinant of a matrix can be thought of as a map det: GLn(R) → R*, where R* is the group R – 0 under multiplication. Since det AB = det A det B, det is a homomorphism. a) Show that the set L of matrices with the form M = (a -b) (b a) where det M ≠ 0 is a subgroup of GL2(R). b) Give a geometric description of the quotient group L/ker(det). (Of course, we're thinking of det as map from L to R*.)

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