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STEP III 1994 question 8 solutionTSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP III 1994 question 8 solution STEP III Question 8
Unparseable or potentially dangerous latex formula. Error 5: Image dimensions are out of bounds: 2056x22
(z_1z_3-z_2z_4^*)(z_1^*z_3^*-z_2^*z_4)-(z_1z_4+z_2z_3^*)(-z_2^*z_3-z_1^*z_4^*)= \newline z_1z_1^*z_3z_3^*-z_1z_3z_2^*z_4-z_2z_4^*z_1^*z_3^*+z_2z_2^*z_4z_4^*+z_1z_4z_2^*z_3+z_1z_1^*z_4z_4^*+z_2z_2^*z_3z_3^*+z_1^*z_2z_3^*z_4^* = \newline z_1z_1^*(z_3z_3^*+z_4z_4^*)+z_2z_2^*(z_3z_3^*+z_4z_4^*)= \newline (z_1z_1^*+z_2z_2^*)(z_3z_3^*+z_4z_4^*)=(|z_1|^2+|z_2|^2)(|z_3|^2+|z_4|^2) (In hindsight it would have been less algebra to use |AB|=|A||B|) Now, chose e.g.
and the determinant of the matrix product will be the desired LHS, i.e.
(Provided Now, to get the RHS of the equality we are to prove, take a look at the matrix This matrix also has a real determinant (i.e.
Unparseable or potentially dangerous latex formula. Error 5: Image dimensions are out of bounds: 1077x24
L_1=\frac{1}{2}((p+qi)(a+bi)-(r+si)(c-di)+(p-qi)(a-bi)-(r-si)(c+di))=\frac{1}{2}(2ap-2rc-2sd-2bq)=ap-bq-cr-ds
So
We also have Therefore follows that Unparseable or potentially dangerous latex formula. Error 5: Image dimensions are out of bounds: 1069x24
L_3=\frac{1}{2}((p+qi)(c+di)+(r+si)(a-bi)+(p-qi)(c-di)+(r-si)(a+bi)=\frac{1}{2}(2pc+2ar+2bs-2qd)=pc+ar+bs-qd
So
Solution by nota bene |
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