The element in the i row j column is moved to j row i column.
For simplicity, you can imagine that the determinant is flipped along the principal diagonal line (which is the top-left to bottom-right one).
Example:Even if it isn't square matrix it still works:
In HKDSE, transpose is almost useless so I won't discuss its main properties, except one:
It may help when we can't solve a determinants, flipping it gives more clues. Take the HW as example:
Example: Factorize
Now notice that when
Therefore,
(the factorization has been demonstrated before) which gives the answer -210.
Note that transpose has given the symmetricities between rows and columns.
Uniqueness of degree in determinants
Sometimes the factorization does not give a unique degree in the polynomial expression which may cause difficulties in factorization.
Recall that determinants is the sum of product of elements from distinct rows and columns. Therefore it each row or column has unique degree, the factorization result has a unique degree.
For example,
In such case you can treat 1 as a variable in order to equate the degree. The determinant becomes:
Note that this determinant is NOT cyclic because each row are not in cyclic relationship.
Cyclic group of (a,b,c): (a,b,c) -> (b,c,a) -> (c,a,b) -> (a,b,c)
Cyclic group of (b,a,c): (b,a,c) -> (a,c,b) -> (c,b,a) -> (b,a,c)
They are two different group, and in this example, (c,a,b) can't reach (a,c,b) by cyclic relationship.
However we see that (a,b) is symmetrical because
Therefore if
Notice that when
Though it's not cyclic, each row still contain a,b,c, which tell us (a+b+c) is a factor of it (this is an usual and important practice in factorizing determinants).
Equating degree,
Case for x=y=z
When two equal variables "kill" the determinant, we say (a-b) is a factor. How to determine the factor when the condition "x=y=z" kill the determinant?
If x=y=z is the condition to make determinant zero, then
Prove:
This is a simplified version of AM-GM inequalities or Rearrangement Inequalities, HKALE players should be familiar with it.
Example:
Factorize
Step 0: Notice that the determinant is cyclic.
Step 1: Notice that the determinant vanishes when x=y=z, so
Step 2: Equating degree, the degree of determinant is 3 while the current factor is 2, that remains one more factor with degree 1, then that must be (a+b+c).
Step 3: Equating coefficient,
Step 4: The second part of the question:
Therefore
For ugly result which can't be factorized
In this case we can't do much other than directly expansion.
Example: express
Notice that
1) This is symmetrical (as we have proven)
2) Each term of the polynomal does not contain a term with a square or a cube of a certain variable like
Then we can conclude that
By equating the coefficient, we have
Example: express
Notice that:
Exercise:
1)Show that for distinct a,b,c,
2)Factorize
3)Factorize
4)Show that (a,c) is symmetrical for
5)Show that
6)Why
Extra question:
For a square matrix
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