I have read the proof for finding the determinant of a 2x2 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 3x3 matrix, all the sources that I have read purely state that the determinant of a 3x3 matrix defined as a formula (omitted here, basically it's summing up the entry of a row/column * determinant of a 2x2 matrix). However, unlike the 2x2 matrix determinant formula, no proof is given.

Similarly, the formula for the determinant of an nxn 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 nxn matrix.

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.