Determinant of Order 3

4.1.2 Determinant of Order 3

Definition. A determinant of order 3 is a square arrangement of 9 elements enclosed between two vertical bars, arranged in 3 rows and 3 columns:

Here $a_{ij}$ denotes the element in the $i$-th row and $j$-th column. For instance $a_{31}$ is the element in the 3rd row, 1st column. A determinant is usually named by a capital letter, or by $\Delta$ (delta). The rows are $R_1,R_2,R_3$ and the columns $C_1,C_2,C_3$ — e.g. the 2nd row is $[a_{21}\ \ a_{22}\ \ a_{23}]$ and the 3rd column is $\begin{bmatrix}a_{13}\\a_{23}\\a_{33}\end{bmatrix}$.

Expansion of a determinant

A 3×3 determinant can be expanded along any of its 3 rows or 3 columns — six expansions in all, and every one gives the same value (this is proved as Property 1 in §4.2). Expanding along the first row:

Notice the alternating $+,-,+$ signs, and that each 2×2 determinant is obtained by deleting the row and column of the multiplying element (this "leftover" 2×2 determinant is formally called a minor, defined in §4.1.3).

Worked Examples

Example (i). Evaluate $\begin{vmatrix} 3 & -4 & 5 \\ -1 & -1 & 2 \\ -2 & -3 & 1 \end{vmatrix}$.

Step 1: Expand along row 1: $D = 3\begin{vmatrix}-1&2\\-3&1\end{vmatrix} - (-4)\begin{vmatrix}-1&2\\-2&1\end{vmatrix} + 5\begin{vmatrix}-1&-1\\-2&-3\end{vmatrix}$.

Step 2: Evaluate each 2×2 piece: $\begin{vmatrix}-1&2\\-3&1\end{vmatrix}=(-1)(1)-(2)(-3)=-1+6=5$; $\begin{vmatrix}-1&2\\-2&1\end{vmatrix}=(-1)(1)-(2)(-2)=-1+4=3$; $\begin{vmatrix}-1&-1\\-2&-3\end{vmatrix}=(-1)(-3)-(-1)(-2)=3-2=1$.

Step 3: Combine: $D=3(5)+4(3)+5(1)=15+12+5$.

Example (ii). Evaluate $\begin{vmatrix} \sec\theta & \tan\theta & 0 \\ \tan\theta & \sec\theta & 0 \\ 0 & 0 & 1 \end{vmatrix}$.

Step 1: The third column has only one non-zero entry ($1$ in row 3), so expand along the third column: $D = 1\times\begin{vmatrix}\sec\theta & \tan\theta\\ \tan\theta & \sec\theta\end{vmatrix}$ (the sign is $(-1)^{3+3}=+1$).

Step 2: Evaluate the 2×2 piece: $\sec^2\theta - \tan^2\theta$.

Step 3: Use the identity $\sec^2\theta-\tan^2\theta=1$.

Example (iii). Evaluate $\begin{vmatrix} 2-i & 3 & 1 \\ 3 & 2-i & 0 \\ -1 & -2 & 2-i \end{vmatrix}$, where $i=\sqrt{-1}$. …