Properties of Determinants

Why Properties Matter

Expanding a $3\times3$ determinant directly, entry by entry, becomes tedious once the entries are large or algebraic. The properties below let a determinant be simplified first -- often reducing it to a form with a whole row or column of zeros, or an easily-spotted value of $0$ -- before any expansion is attempted. Every property stated here holds for determinants of any order, but is illustrated for $2\times2$ or $3\times3$ to match this chapter's scope. Throughout, $R_i$ denotes the $i$th row and $C_i$ the $i$th column.

Property 1 -- Invariance Under Transpose

$|A| = |A^{T}|$: the determinant of a matrix equals the determinant of its transpose. Consequently, every property stated below for rows applies equally to columns, and vice versa.

Property 2 -- Interchanging Two Rows (or Columns) Reverses the Sign

If any two rows (or any two columns) of $A$ are interchanged, the determinant of the resulting matrix is $-|A|$. For example, $\begin{vmatrix}2&5\\1&3\end{vmatrix}=1$, while interchanging the rows gives $\begin{vmatrix}1&3\\2&5\end{vmatrix}=5-6=-1$, exactly the negative.

Property 3 -- Two Identical Rows (or Columns) Give a Zero Determinant

If any two rows (or any two columns) of $A$ are identical, then $|A|=0$. This follows immediately from Property 2: swapping the two identical rows must both leave the matrix unchanged (since they are equal) and negate the determinant, and the only number equal to its own negative is $0$.

Property 4 -- A Common Factor Can Be Taken Out

If every entry of one row (or one column) of $A$ is multiplied by a scalar $k$, the determinant of the new matrix is $k|A|$. Equivalently, a common factor of every entry in a single row or column may be pulled outside the determinant sign. For example, $\begin{vmatrix}6&9\\4&6\end{vmatrix}$ has every entry of row 1 divisible by $3$: $=3\begin{vmatrix}2&3\\4&6\end{vmatrix}$; and here every entry of the new row 2 is divisible by $2$ as well, so $=3\times2\begin{vmatrix}2&3\\2&3\end{vmatrix}=6\times0=0$ by Property 3, since the two rows have become identical.

Property 5 -- Sum Property

If every entry of one row (say $R_1$) of $A$ is written as a sum of two parts, then $|A|$ splits as the sum of two determinants, each having that row replaced by one of the two parts and every other row unchanged. In symbols, if $R_1=R_1'+R_1''$, then $|A|=|A \text{ with } R_1'|+|A \text{ with } R_1''|$.

Property 6 -- Invariance Under a Row/Column Combination (the Most Useful Property)

If a scalar multiple of one row (or column) is added to another row (or column), the determinant is unchanged. In symbols, replacing $R_i$ by $R_i+kR_j$ (for $i\ne j$) leaves $|A|$ the same. This is the single most powerful simplification tool in the chapter: it is routinely used to create zeros in a row or column (so that Section 1's "expand along the row with the most zeros" tip becomes free), or to reveal that two rows have become proportional -- which forces the determinant to $0$ by Properties 3 and 4 combined. …