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STEP III 1994 question 8 solution
From The Student RoomTSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP III 1994 question 8 solution STEP III Question 8
(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.
So
So
We also have Therefore follows that
So
Solution by nota bene |











The determinant of this matrix is
.
, because then there is a zero-determinant, which we don't want)
again.
You can see that if we let
and
we have:
'
).
Now, if we set
and
this means the matrix has the determinant
, which is the RHS in the equality.
,
,
and
and
and





