Determinants 3.5

This is when we investigate how we find the formulae for the determinants of matrixes of bigger size.

We remember that the formulas for the determinants are \(a_{11}a_{22}-a_{12}a_{21}\) (for a 2 by 2 matrix) and \(a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{12}a_{21}a_{33}-a_{11}a_{23}a_{32}-a_{13}a_{22}a_{32}\) (for a 3 by 3 matrix)

First, we take each entry as \(a_{mn}\)

FYI, I am writing this as a note as I do not know how to put it in questions.

  1. Look at the first number of each entry. I mean the '123' in each \(a_{11}a_{22}a_{33}\) We take out the first number (m) of each 'a'. They are all consecutive.

  2. Now, we look at the second number (n). They all have numbers 1, 2, and 3 in total, but jumbled up in all combinations: 123, 132, 213, 231, 312, 321.

  3. Now, we lastly have to see what sign follows which entry. We will look at the 'n'. 123, 231 and 312 have a positive sign to it. 321, 213 and 132 have a negative sign to it. We take each set of three numbers and look at how many times you have to swap one number with another to get back '123...' in the least number of swaps. If it is even, that entry has a positive sign. If it is odd, that entry has a negative sign. Here's some examples:

123 (0)

231-->132-->123(2)

312-->321-->123(2)

However,

321-->123(1)

213-->123(1)

132-->123(1)

With these three rules, we can determine the determinant for any square matrix.

From this note onwards, I will try to find patterns in some matrixes. So in the case you don't understand yet, you can try 4 (without guess1ng).

Note by Aloysius Ng
3 years, 4 months ago

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