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Estimated time: 6 minutes
CBSE: Class 12
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.
CBSE: Class 12
Formula: Expansion
For a matrix A, expanding along the first row (\[R_1\]) looks like this:
\[|A| = (-1)^{1+1} a_{11} \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} + (-1)^{1+2} a_{12} \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + (-1)^{1+3} a_{13} \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}\]
Invariance: Expanding the determinant along any row (e.g., \[R_1, R_2, R_3\]) or any column (\[C_1, C_2, C_3\]) will always yield the exact same final value.
CBSE: Class 12
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
\[\Delta = 4 \begin{vmatrix} -1 & 3 \\ 4 & 1 \end{vmatrix} - 0 \begin{vmatrix} 1 & 2 \\ 4 & 1 \end{vmatrix} + 0 \begin{vmatrix} 1 & 2 \\ -1 & 3 \end{vmatrix}\]
\[= 4 (-1 - 12) - 0 + 0 = - 52\]
CBSE: Class 12
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 |
