Determinants
Inverse, Adjoint, and Systems of Equations
1. Inverse of a Matrix ($$A^{-1}$$)
| Definition / Property | Formula / Condition |
|---|---|
| Standard Formula | $$A^{-1} = \frac{1}{|A|} \text{adj}(A)$$ |
| Existence Condition | $$|A| \neq 0$$ (Non-singular) |
| Reversal Law | $$(AB)^{-1} = B^{-1} A^{-1}$$ |
| Transpose Property | $$(A^T)^{-1} = (A^{-1})^T$$ |
| Determinant of Inverse | $$|A^{-1}| = \frac{1}{|A|}$$ |
2. Properties of Adjoint ($$\text{adj } A$$)
| Fundamental Property | $$A(\text{adj } A) = (\text{adj } A)A = |A|I$$ |
| Determinant of Adjoint | $$|\text{adj } A| = |A|^{n-1}$$ |
| Adjoint of Product | $$\text{adj}(AB) = (\text{adj } B)(\text{adj } A)$$ |
| Adjoint of Adjoint | $$\text{adj}(\text{adj } A) = |A|^{n-2} A$$ |
| Det of $$A(\text{adj } A)$$ | $$|A(\text{adj } A)| = |A|^n$$ |
3. Matrix Method Solutions ($$AX = B$$)
| Condition | Solution Type |
|---|---|
| $$|A| \neq 0$$ | Unique Solution ($$X = A^{-1}B$$) |
| $$|A| = 0, (\text{adj } A)B \neq O$$ | No Solution (Inconsistent) |
| $$|A| = 0, (\text{adj } A)B = O$$ | Infinite Solutions (Consistent) |
4. Properties of Cofactors ($$A_{ij}$$)
| Sum with same row elements | $$a_{i1}A_{i1} + a_{i2}A_{i2} + a_{i3}A_{i3} = |A|$$ |
| Sum with other row elements | $$a_{i1}A_{j1} + a_{i2}A_{j2} + a_{i3}A_{j3} = 0 \text{ (for } i \neq j)$$ |
5. Area of Triangle using Determinants
$$\text{Area} = \left| \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|$$
*Points are collinear if Area = 0.