I have read the proof for finding the determinant of a $2 \times 2$ matrix.
It makes sense, since for a matrix
\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
$(ad-bc)$ must be non-zero for the inverse of the matrix to exist. So it is logical that $(ad-bc)$ is the determinant.
However when it comes to a $3 \times 3$ matrix, all the sources that I have read purely state that the determinant of a $3 \times 3$ matrix defined as a formula (omitted here, basically it's summing up the entry of a row/column * determinant of a $2 \times 2$ matrix). However, unlike the $2 \times 2$ matrix determinant formula, no proof is given.
Similarly, the formula for the determinant of an $n \times n$ matrix is not given in my textbook. Unfortunately, I can't seem to find a proof that I could comprehend on the internet. It would be great if someone can give me a proof of the formula for finding the determinant of an $n \times n$ matrix.
Best Answer
This is the Definition of Determinant