3x3 matrix operations intuition help

Question

So I understand the intuition of taking the determinant of a 2x2 matrix, but what is the intuition for taking the determinant of 3x3, matrix? It makes zero intuitive sense just looking at it. Also, when finding the inverse of a matrix, why do we need to find the cofactor matrix and the adjugate matrix and transpose the cofactor matrix, and what is the point of the checkerboard matrix with + and - signs?

Answer 1

Suppose we have a matrix $A=(a_{ij})_{ij}$. Lets denote the determinant of $A$ once we remove the $i$th row and the $j$th column as $A_{i,j}$ and lets denote $(-1)^{i+j}\cdot A_{i,j}$ as $C_{i,j}$ (the $i,j$ cofactor of $A$). Then, by definition, the determinant of a matrix of dimension 1x1 is the number itself and for any other dimension of $A$ is:

$$|A| = a_{11}C_{11} + a_{12}C_{12} + \dots + a_{1n}C_{1n}$$

(By Laplace extension's Theorem you can use any row not necessarily the first one)

that means that

$$\begin{vmatrix}a&b\\c&d\end{vmatrix} = a \begin{vmatrix}d\end{vmatrix} - b\begin{vmatrix}c\end{vmatrix} = ad-bc$$

And in the case of a 3x3 matrix

$$\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix} = a\begin{vmatrix}e&f\\h&i\end{vmatrix}-b\begin{vmatrix}d&f\\g&i\end{vmatrix}+c\begin{vmatrix}d&e\\g&h\end{vmatrix} = aei-afh-bdi+bgf+cdh-cge$$

So even though at first glance the determinant of a 3x3 matrix it may seem a non intuitive result, it actually is a result of computing determinants 2x2.