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Definition: Expansion Method
To find the determinant, multiply each element of your chosen row (or column) by its corresponding sign multiplier \[(-1)^{i+j}\] and the \[2 \times 2\] determinant that remains after deleting that element's row and column.
Formula: Expansion
To expand along the first row, multiply each element of the first row by: \[ (-1)^{i+j} \] and by the second-order determinant obtained after deleting the row and column containing that element.
Thus,
\[ |A| = (-1)^{1+1}a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} + (-1)^{1+2}a_{12}\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + (-1)^{1+3}a_{13}\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}. \]
Since the signs are \[+, -, +\],
\[ |A| = a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12}\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13}\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix} \]
or,
\[ |A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]
Expansion Along the Second Row \[(R_2)\]
The sign pattern along the second row is \[ -, +, -. \]
Therefore,
\[ |A| = -a_{21}\begin{vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{vmatrix} + a_{22}\begin{vmatrix} a_{11} & a_{13} \\ a_{31} & a_{33} \end{vmatrix} - a_{23}\begin{vmatrix} a_{11} & a_{12} \\ a_{31} & a_{32} \end{vmatrix}. \]
Expansion along \[R_2\] gives the same value as expansion along \[R_1\].
Expansion Along the First Column \[(C_1)\]
The signs down the first column are \[ +, -, +. \]
Thus,
\[ |A| = a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{21}\begin{vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{vmatrix} + a_{31}\begin{vmatrix} a_{12} & a_{13} \\ a_{22} & a_{23} \end{vmatrix}. \]
Again, the final value is the same.
Example 1
Evaluate the determinant \[\Delta = \begin{vmatrix} 1 & 2 & 4 \\ -1 & 3 & 0 \\ 4 & 1 & 0 \end{vmatrix}\].
Solution:
Note that in the third column, two entries are zero. So expanding along third column (\[\text{C}_{3}\]), we get
Key Points: Expansion of Determinant
| Concept | Formula/Rule |
|---|---|
| Expansion along R₁ | a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ |
| Expansion along C₁ | a₁₁C₁₁ + a₂₁C₂₁ + a₃₁C₃₁ |
| Cofactor Sign | (-1)(i+j) → checkerboard: + - + / - + - / + - + |
| Zero Strategy | Expand along row/column with most zeros |
| Result Independence | Any row/column expansion gives same |
| Important Result | Order 3 determinant: 6 expansions (R1,R2,R3,C1,C2,C3),all give the same value |
