Determinants
Har square matrix ke saath ek number juda hota hai jise determinant kehte hain. Ye batata hai ki matrix invertible hai ya nahi, aur linear equations ko solve karne me kaam aata hai.
1. Determinant of \(2 \times 2\)
\[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \qquad |A| = ad - bc. \]
\[ \begin{vmatrix} 3 & 1 \\ 2 & 4 \end{vmatrix} = (3)(4) - (1)(2) = 12 - 2 = 10. \]
2. Determinant of \(3 \times 3\)
First row ke saath expand karte hain (cofactor expansion):
\[ \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = a_1\begin{vmatrix} b_2 & c_2 \\ b_3 & c_3 \end{vmatrix} - b_1\begin{vmatrix} a_2 & c_2 \\ a_3 & c_3 \end{vmatrix} + c_1\begin{vmatrix} a_2 & b_2 \\ a_3 & b_3 \end{vmatrix}. \]
(\(+, -, +\) ka sign pattern dhyan rakho.)
3. Minors aur Cofactors
- Minor \(M_{ij}\): element \(a_{ij}\) ki row aur column hata kar bachi matrix ka determinant.
- Cofactor \(A_{ij} = (-1)^{i+j} M_{ij}\).
4. Properties of Determinants
- Do rows (ya columns) aapas me badlo → determinant ka sign palat jaata hai.
- Do rows/columns same ho → determinant \(= 0\).
- Kisi ek row/column ko \(k\) se multiply karo → determinant \(k\) guna.
- \(|A^{T}| = |A|\), aur \(|AB| = |A|\,|B|\).
5. Adjoint aur Inverse
Adjoint \(\operatorname{adj}(A)\) = cofactor matrix ka transpose. Inverse: \[ A^{-1} = \frac{1}{|A|}\,\operatorname{adj}(A), \qquad |A| \neq 0. \]
Agar \(|A| = 0\) ho to matrix singular hai aur uska inverse nahi hota.
6. Area of a Triangle
Vertices \((x_1, y_1), (x_2, y_2), (x_3, y_3)\) wale triangle ka area:
\[ \text{Area} = \frac{1}{2}\left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|. \]
Agar ye determinant \(0\) ho, to teeno points ek hi line par (collinear) hain.
7. Solving Linear Equations (Matrix Method)
System \(AX = B\) ko aise solve karte hain:
\[ X = A^{-1}B, \qquad \text{jab } |A| \neq 0. \]
\(|A| \neq 0\) ho to unique solution milta hai (consistent system).
\(|A| = \begin{vmatrix} 2 & 3 \\ 1 & 2 \end{vmatrix} = 1 \neq 0\), toh \(A\) invertible hai aur \[ A^{-1} = \frac{1}{1}\begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}. \]
Key Takeaways
- \(2\times2\): \(|A| = ad - bc\); \(3\times3\) cofactor expansion se.
- Cofactor \(A_{ij} = (-1)^{i+j}M_{ij}\).
- \(A^{-1} = \frac{1}{|A|}\operatorname{adj}(A)\); \(|A| = 0\) → singular, koi inverse nahi.
- Area of triangle aur collinearity determinant se.
- \(AX = B \Rightarrow X = A^{-1}B\) (jab \(|A| \neq 0\)).