Properties of Determinants

Property 1 (transpose). $|A^T|=|A|$ — since expanding by rows gives the same value as expanding by columns.

Property 2 (row/column swap). Interchanging any two rows (or columns) flips the sign of the determinant, leaving its absolute value unchanged.

Property 3 (repeated swaps). $n$ successive row/column interchanges multiply the determinant by $(-1)^n$.

Property 4 (identical rows/columns). If two rows (or two columns) of $A$ are identical, $|A|=0$. (Reasoning: swapping those two identical rows leaves the matrix looking the same, yet Property 2 says the swap must flip the sign — so $|A|=-|A|$, forcing $|A|=0$.)

Property 5 (proportional rows/columns). If one row (or column) is a scalar multiple of another, $|A|=0$.

Property 6 (scalar factor). Multiplying every entry of one row (or column) by a scalar $k$ multiplies the whole determinant by $k$. In particular, for an $n\times n$ matrix $A$, $|kA|=k^n|A|$ (every one of the $n$ rows carries the factor $k$).

Property 7 (splitting a sum). If every entry of one row (or column) is itself a sum of two terms, e.g. $\begin{vmatrix}a_1+m_1&b_1&c_1\\a_2+m_2&b_2&c_2\\a_3+m_3&b_3&c_3\end{vmatrix}$, the determinant splits into the sum of two determinants — one with the $a$'s in that column, one with the $m$'s, everything else unchanged.

Property 8 (invariant row/column operations). Adding to any row (or column) a scalar multiple of one or more other rows (or columns) — e.g. $R_1\to R_1+pR_2+qR_3$ — leaves the determinant unchanged. This is the single most useful trick for creating zeros before expanding. …