Determinants

1. For easier calculations, we shall expand the determinant along that row or column which contains maximum number of zeros.
2. While expanding, instead of multiplying by \((-1)^{i+j}\), we can multiply by +1 or-1 according as \((i+j)\) is even or odd.
3. Let \(A=[\begin{smallmatrix}2&2\\ 4&0\end{smallmatrix}]\) and \(B=[\begin{smallmatrix}1&1\\ 2&0\end{smallmatrix}]\) Then, it is easy to verify that \(A=2B\) Also \(|A|=0-8=-8\) and \(|B|=0-2=-2.\) Observe that, \(|A|=4(-2)=2^{2}|B|\) or \(|A|=2^{n}|B|\) , where \(n=2\) is the order of square matrices A and B.

In general, if \(A=kB\) where A and B are square matrices of order n, then | \(A|=k^{n}\) | B , where \(n=1,2,3\)