Properties of Determinants

4.2 Properties of Determinants

Expanding a large determinant term-by-term is slow and error-prone. The properties below let us simplify a determinant — using row/column operations — before we ever expand it.

Notation. $R_i \leftrightarrow R_j$ means "interchange rows $i$ and $j$"; $C_i \leftrightarrow C_j$ means the same for columns; $R_i \to R_i + kR_j$ means "replace row $i$ by row $i$ plus $k$ times row $j$".

Property 1 (Transpose invariance). The value of a determinant is unchanged if its rows are turned into columns and its columns into rows.

Both sides expand, by direct cofactor expansion, to the same six-term sum $a_1(b_2c_3-b_3c_2)-b_1(a_2c_3-a_3c_2)+c_1(a_2b_3-a_3b_2)$ — this is exactly why a determinant can be expanded along ANY row or column with the same result.

Property 2 (Row/column swap flips the sign). If any two rows (or columns) are interchanged, the value of the determinant changes sign: $R_i\leftrightarrow R_j$ turns $D$ into $-D$.

Property 3 (Repeated row/column ⇒ zero). If two rows (or columns) of a determinant are identical, its value is $0$. Proof: if $R_1=R_2$, swapping $R_1\leftrightarrow R_2$ gives $D_1=-D$ (Property 2), but since the rows were identical the swap changes nothing, so $D_1=D$. Adding these, $2D=0$, so $D=0$.

Property 4 (Scalar factor of a row/column). If every element of one row (or column) is multiplied by a constant $k$, the new determinant is $k$ times the original: $R_i\to kR_i$ multiplies $D$ by $k$. Two consequences: (i) a common factor in any single row or column can be pulled outside the determinant; (ii) if two rows (or columns) are proportional, the determinant is $0$ (pull out the proportionality constant, leaving two identical rows — Property 3).

Property 5 (Splitting a row into a sum). If every element of one row (or column) is itself a sum of two numbers, the determinant splits into a sum of two determinants:

Property 6 (Adding a multiple of one row to another). If a constant multiple of one row (or column) is added to another row (or column), the value of the determinant is unchanged: $R_i\to R_i+kR_j$ leaves $D$ unchanged. This is the workhorse property for simplifying a determinant before expanding it — it lets you create zeros strategically.

Main diagonal. For a determinant $A=[a_{ij}]$, the elements $a_{11},a_{22},a_{33},\dots,a_{nn}$ (where $i=j$) are its main (principal) diagonal.

Property 7 (Triangle property). If every element above OR below the main diagonal is zero, the value of the determinant equals the product of its diagonal elements:

Remark: if an entire row or an entire column is all zeros, the determinant is $0$ (expand along that row/column — every term vanishes).

Worked Examples

Example 1. Show that (i) $\begin{vmatrix}101&202&303\\505&606&707\\1&2&3\end{vmatrix}=0$ and (ii) $\begin{vmatrix}312&313&314\\315&316&317\\318&319&320\end{vmatrix}=0$.

1. Step 1: $R_1\to R_1-R_3$ turns row 1 into $(100,200,300)=100(1,2,3)$, matching row 3 exactly (up to the factor 100). Step 2: pull out 100 (Property 4); now rows 1 and 3 are identical, so by Property 3 the determinant is $0$.
2. Step 1: $C_2\to C_2-C_1$ and $C_3\to C_3-C_1$ turn every row into $(\text{base},1,2)$ — because consecutive integers differ by 1 and 2. Step 2: pull out the common factor $2$ from column 3 (Property 4); columns 2 and 3 are now identical ($1,1,1$ pattern after scaling), so by Property 3 the value is $0$. Example 2. Prove $\begin{vmatrix}1&1&1\\a&b&c\\bc&ca&ab\end{vmatrix}=\begin{vmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{vmatrix}$. …