Determinants – Complete Revision Series (JEE Level)
Determinants form the backbone of solving linear equations, understanding area, geometry, and matrix algebra. This chapter is formula-rich but logic-driven — once patterns are clear, questions become mechanical.
1. Determinant of Order 2
For a matrix \[ A=\begin{vmatrix} a & b\\ c & d \end{vmatrix} \]
\[ |A| = ad - bc \]
Important Results:
- \(|A| = 0\) ⇒ rows/columns are linearly dependent
- Interchanging rows changes sign of determinant
- Multiplying a row by \(k\) multiplies determinant by \(k\)
2. Determinant of Order 3
\[ \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix} = a(ei-fh)-b(di-fg)+c(dh-eg) \]
Student Pain Point:
Sign mistakes occur due to improper expansion. Always expand along a row/column with
maximum zeros.
3. Minors & Cofactors
Minor of \(a_{ij}\) = determinant obtained by deleting \(i^{th}\) row and \(j^{th}\) column
\[ C_{ij} = (-1)^{i+j} M_{ij} \]
Key Identity:
\[ |A| = \sum a_{ij} C_{ij} \]4. Properties of Determinants
- If two rows/columns are identical ⇒ determinant = 0
- If one row is proportional to another ⇒ determinant = 0
- \(|A^T| = |A|\)
- If elements of a row are sum of two rows ⇒ determinant splits linearly
Golden Rule: Apply row/column operations to simplify BEFORE expanding.
5. Area Using Determinants
Area of triangle with vertices \((x_1,y_1),(x_2,y_2),(x_3,y_3)\)
\[ \text{Area} = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1 \end{vmatrix} \right| \]Collinearity Condition: Area = 0
6. Cramer’s Rule (System of Linear Equations)
For system: \[ a_1x+b_1y+c_1z=d_1\\ a_2x+b_2y+c_2z=d_2\\ a_3x+b_3y+c_3z=d_3 \]
\[ x=\frac{\Delta_x}{\Delta},\quad y=\frac{\Delta_y}{\Delta},\quad z=\frac{\Delta_z}{\Delta} \]Conditions:
- \(\Delta \neq 0\) ⇒ unique solution
- \(\Delta = 0,\ \Delta_x=\Delta_y=\Delta_z=0\) ⇒ infinite solutions
- \(\Delta = 0,\) any numerator ≠ 0 ⇒ no solution
7. Determinant Tricks (JEE Focus)
- Use row/column subtraction to create zeros
- Take common terms outside determinant
- Convert determinant into triangular form
- Symmetric determinants often factorize
Exam Insight: Most JEE problems test properties, not raw expansion.
8. Link with Other Chapters
- Matrices: Inverse exists iff determinant ≠ 0
- Coordinate Geometry: Area, collinearity
- Linear Equations: Cramer’s Rule
Part of StudyBeacon Revision Series – Learn smart, revise sharper.
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