Use the relation | A B | = | A | | B | to show that ( a 1 2 + a 2 2 ) ( b 1 2 + b 2 2 ) = ( a 1 b 1 − a 2 b 2 ) 2 + ( a 2 b 1 + a 1 b 2 ) 2 .

Question

Use the relation       |    A  B      |    =      |    A      |        |    B      |   to show that  (      a    1    2    +      a    2    2    )  (      b    1    2    +      b    2    2    )  =  (      a    1        b    1    −      a    2        b    2        )    2    +  (      a    2        b    1    +      a    1        b    2        )    2    .

Jaden Mason · Accepted Answer

I believe       |    A      |   is used to denote the determinant of A in this case. The matrices you are looking for are given by  A  =      (                                        a            1                                                a            2                                                −                      a            2                                                a            1                                )    ,    B  =      (                                        b            1                                                b            2                                                −                      b            2                                                b            1                                )    .Using       |    A  B      |    =      |    A      |        |    B      |   for these matrices results in the identity you want.This representation of   (      a    1    ,      a    2    ) by A is actually a a way to represent complex numbers using real matrices. Equivalently, you could define       z    a    =      a    1    +      a    2    i, and use the identity       |        z    a        z    b        |    =      |        z    a        |        |        z    b        |  , where       |    z      |   is the norm of the complex number   z  ∈      C

Use the relation |AB|=|A||B| to show that (a_1^2+a_2^2)(b_1^2+b_2^2)=(a_1b_1−a_2b_2)^2+(a_2b_1+a_1b_2)^2.

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