The Laplace expansion over the first row the determinant of an $(n\times n)$ Matrix $A$ is defined as
$det(A)=\sum_{j=1}^{3}(-1)^{1+j}M_{1j}a_{1j}$ where $M_{ij}$ is the determinant of the $(n-1)\times (n-1)$ formed by removing the $i$th row and $jth$ column of A.
I am trying to prove that the expansion gives the same result independent over the choice of row over which it is expanded.
My attempt so far has to be consider a new matrix $B$ created from $A$ by permuting the rows such that the first row of $B$ is the $i$th row of $A$. As row $i$ in A is swapped $(i-1)$ times to become the first row of $B$, we can say that
$det(B)=(-1)^{i-1}det(A)$
I am now trying to reach a result that states
$det(A)=\sum_{j=1}^{3}(-1)^{i+j}M_{ij}a_{ij}$
which would prove that the expansion can be computed along any choice of row $i$ assuming that the first row gives the determinant, im not sure how to get to this point from my matrix $B$