Question

How do I find the determinant of of a `4xx4` matrix?

Answer 1

Recursively in terms of determinants of `3xx3` matrices...

Explanation:

Given a matrix:

`((a_11, a_12, a_13, a_14),(a_21, a_22, a_23, a_24),(a_31, a_32, a_33, a_34), (a_41, a_42, a_43, a_44))`

Duplicate the first three columns to form three extra columns. Then for each of the first `4` terms in the top row, `a_11, a_12, a_13, a_14` multiply by the determinant of the `3xx3` matrix immediately below and to the right. Add these together to get your determinant.

`((color(red)(a_11), a_12, a_13, a_14, a_11, a_12, a_13),(a_21, color(red)(a_22), color(red)(a_23), color(red)(a_24), a_21, a_22, a_23),(a_31, color(red)(a_32), color(red)(a_33), color(red)(a_34), a_31, a_32, a_33), (a_41, color(red)(a_42), color(red)(a_43), color(red)(a_44), a_41, a_42, a_43))`

`((a_11, color(red)(a_12), a_13, a_14, a_11, a_12, a_13),(a_21, a_22, color(red)(a_23), color(red)(a_24), color(red)(a_21), a_22, a_23),(a_31, a_32, color(red)(a_33), color(red)(a_34), color(red)(a_31), a_32, a_33), (a_41, a_42, color(red)(a_43), color(red)(a_44), color(red)(a_41), a_42, a_43))`

`((a_11, a_12, color(red)(a_13), a_14, a_11, a_12, a_13),(a_21, a_22, a_23, color(red)(a_24), color(red)(a_21), color(red)(a_22), a_23),(a_31, a_32, a_33, color(red)(a_34), color(red)(a_31), color(red)(a_32), a_33), (a_41, a_42, a_43, color(red)(a_44), color(red)(a_41), color(red)(a_42), a_43))`

`((a_11, a_12, a_13, color(red)(a_14), a_11, a_12, a_13),(a_21, a_22, a_23, a_24, color(red)(a_21), color(red)(a_22), color(red)(a_23)),(a_31, a_32, a_33, a_34, color(red)(a_31), color(red)(a_32), color(red)(a_33)), (a_41, a_42, a_43, a_44, color(red)(a_41), color(red)(a_42), color(red)(a_43)))`

`abs((a_11, a_12, a_13, a_14),(a_21, a_22, a_23, a_24),(a_31, a_32, a_33, a_34), (a_41, a_42, a_43, a_44))`

`= a_11 abs((a_22, a_23, a_24), (a_32, a_33, a_34), (a_42, a_43, a_44))+ a_12 abs((a_23, a_24, a_21), (a_33, a_34, a_31), (a_43, a_44, a_41))+ a_13 abs((a_24, a_21, a_22), (a_34, a_31, a_32), (a_44, a_41, a_42)) + a_14 abs((a_21, a_22, a_23), (a_31, a_32, a_33), (a_41, a_42, a_43))`

Answer 2

I have two main methods:

Explanation:

You have two main ways to find the determinant of a matrix `4xx4`:

1] Using the method of Laplace or of the Cofactors. This method is good and easy to apply but very cumbersome! You must evaluate a lot of smaller determinants and it is possible, during these steps, to make mistakes (it also quite boring!!!):
For example:

2] The second way is a little bit more fun but a little bit more daring. You must (through operations on lines or columns) get a line or column of all zeroes but one.
The operations you are allowed to make are:
- add or subtract 2 lines or columns;
- multiply the elements of one line or columns times a number and then add or subtract to another line or column. For example:

Extract the number at the crossing (with the right sign) and use it together with the remaining `3xx3` determinant: