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

Jul 27, 2015

Recursively in terms of determinants of $3 \times 3$ matrices...

#### Explanation:

Given a matrix:

$\left(\begin{matrix}{a}_{11} & {a}_{12} & {a}_{13} & {a}_{14} \\ {a}_{21} & {a}_{22} & {a}_{23} & {a}_{24} \\ {a}_{31} & {a}_{32} & {a}_{33} & {a}_{34} \\ {a}_{41} & {a}_{42} & {a}_{43} & {a}_{44}\end{matrix}\right)$

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 $3 \times 3$ matrix immediately below and to the right. Add these together to get your determinant.

$\left(\begin{matrix}\textcolor{red}{{a}_{11}} & {a}_{12} & {a}_{13} & {a}_{14} & {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & \textcolor{red}{{a}_{22}} & \textcolor{red}{{a}_{23}} & \textcolor{red}{{a}_{24}} & {a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & \textcolor{red}{{a}_{32}} & \textcolor{red}{{a}_{33}} & \textcolor{red}{{a}_{34}} & {a}_{31} & {a}_{32} & {a}_{33} \\ {a}_{41} & \textcolor{red}{{a}_{42}} & \textcolor{red}{{a}_{43}} & \textcolor{red}{{a}_{44}} & {a}_{41} & {a}_{42} & {a}_{43}\end{matrix}\right)$

$\left(\begin{matrix}{a}_{11} & \textcolor{red}{{a}_{12}} & {a}_{13} & {a}_{14} & {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & \textcolor{red}{{a}_{23}} & \textcolor{red}{{a}_{24}} & \textcolor{red}{{a}_{21}} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & \textcolor{red}{{a}_{33}} & \textcolor{red}{{a}_{34}} & \textcolor{red}{{a}_{31}} & {a}_{32} & {a}_{33} \\ {a}_{41} & {a}_{42} & \textcolor{red}{{a}_{43}} & \textcolor{red}{{a}_{44}} & \textcolor{red}{{a}_{41}} & {a}_{42} & {a}_{43}\end{matrix}\right)$

$\left(\begin{matrix}{a}_{11} & {a}_{12} & \textcolor{red}{{a}_{13}} & {a}_{14} & {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} & \textcolor{red}{{a}_{24}} & \textcolor{red}{{a}_{21}} & \textcolor{red}{{a}_{22}} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33} & \textcolor{red}{{a}_{34}} & \textcolor{red}{{a}_{31}} & \textcolor{red}{{a}_{32}} & {a}_{33} \\ {a}_{41} & {a}_{42} & {a}_{43} & \textcolor{red}{{a}_{44}} & \textcolor{red}{{a}_{41}} & \textcolor{red}{{a}_{42}} & {a}_{43}\end{matrix}\right)$

$\left(\begin{matrix}{a}_{11} & {a}_{12} & {a}_{13} & \textcolor{red}{{a}_{14}} & {a}_{11} & {a}_{12} & {a}_{13} \\ {a}_{21} & {a}_{22} & {a}_{23} & {a}_{24} & \textcolor{red}{{a}_{21}} & \textcolor{red}{{a}_{22}} & \textcolor{red}{{a}_{23}} \\ {a}_{31} & {a}_{32} & {a}_{33} & {a}_{34} & \textcolor{red}{{a}_{31}} & \textcolor{red}{{a}_{32}} & \textcolor{red}{{a}_{33}} \\ {a}_{41} & {a}_{42} & {a}_{43} & {a}_{44} & \textcolor{red}{{a}_{41}} & \textcolor{red}{{a}_{42}} & \textcolor{red}{{a}_{43}}\end{matrix}\right)$

$\left\mid \begin{matrix}{a}_{11} & {a}_{12} & {a}_{13} & {a}_{14} \\ {a}_{21} & {a}_{22} & {a}_{23} & {a}_{24} \\ {a}_{31} & {a}_{32} & {a}_{33} & {a}_{34} \\ {a}_{41} & {a}_{42} & {a}_{43} & {a}_{44}\end{matrix} \right\mid$

$= {a}_{11} \left\mid \begin{matrix}{a}_{22} & {a}_{23} & {a}_{24} \\ {a}_{32} & {a}_{33} & {a}_{34} \\ {a}_{42} & {a}_{43} & {a}_{44}\end{matrix} \right\mid + {a}_{12} \left\mid \begin{matrix}{a}_{23} & {a}_{24} & {a}_{21} \\ {a}_{33} & {a}_{34} & {a}_{31} \\ {a}_{43} & {a}_{44} & {a}_{41}\end{matrix} \right\mid + {a}_{13} \left\mid \begin{matrix}{a}_{24} & {a}_{21} & {a}_{22} \\ {a}_{34} & {a}_{31} & {a}_{32} \\ {a}_{44} & {a}_{41} & {a}_{42}\end{matrix} \right\mid + {a}_{14} \left\mid \begin{matrix}{a}_{21} & {a}_{22} & {a}_{23} \\ {a}_{31} & {a}_{32} & {a}_{33} \\ {a}_{41} & {a}_{42} & {a}_{43}\end{matrix} \right\mid$

Jul 27, 2015

I have two main methods:

#### Explanation:

You have two main ways to find the determinant of a matrix $4 \times 4$:

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 $3 \times 3$ determinant: