📌 1. Introduction to Determinants

A determinant is a scalar value associated with a square matrix. It provides important information about the matrix, such as whether it is invertible or not.

Determinants are denoted by vertical bars.

Example:$$\begin{vmatrix} a & b \\ c & d \end{vmatrix}$$​ac​bd​​

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📖 2. Determinant of a Matrix

🔹 For a 2 × 2 Matrix

$$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad – bc$$​ac​bd​​=ad−bc

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$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad – bc$​ac​bd​​=ad−bc

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🔹 For a 3 × 3 Matrix

$$\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$$​adg​beh​cfi​​

Using expansion:$$= a(ei – fh) – b(di – fg) + c(dh – eg)$$=a(ei−fh)−b(di−fg)+c(dh−eg)

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📐 3. Minors and Cofactors

🔹 Minor

The minor of an element is the determinant obtained after removing its row and column.

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🔹 Cofactor

$$C_{ij} = (-1)^{i+j} \cdot M_{ij}$$Cij​=(−1)i+j⋅Mij​

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$C_{ij} = (-1)^{i+j} M_{ij}$Cij​=(−1)i+jMij​

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🔁 4. Expansion of Determinant

A determinant can be expanded along any row or column using cofactors.$$\Delta = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$$Δ=a11​C11​+a12​C12​+a13​C13​

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📊 5. Properties of Determinants

1. Interchange of Rows

If two rows are interchanged → determinant changes sign.

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2. Two Equal Rows

If two rows are identical → determinant = 0

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3. Zero Row

If any row is zero → determinant = 0

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4. Factor Property

If all elements of a row have a common factor → factor can be taken out

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5. Adding Rows

Adding a multiple of one row to another does not change determinant

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🔄 6. Area Using Determinants

The area of a triangle with vertices:

$(x_1, y_1), (x_2, y_2), (x_3, y_3)$(x1​,y1​),(x2​,y2​),(x3​,y3​)$$\text{Area} = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$$Area=21​​x1​x2​x3​​y1​y2​y3​​111​​

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$\text{Area} = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$Area=21​​x1​x2​x3​​y1​y2​y3​​111​​

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📌 Collinearity Condition

Points are collinear if determinant = 0

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🔑 7. Adjoint of a Matrix

The adjoint of a matrix is the transpose of its cofactor matrix.$$\text{adj}(A) = (C_{ij})^T$$adj(A)=(Cij​)T

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$\text{adj}(A) = (C_{ij})^T$adj(A)=(Cij​)T

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🔁 8. Inverse of a Matrix Using Determinant

$$A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A)$$A−1=∣A∣1​⋅adj(A)

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$A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A)$A−1=∣A∣1​⋅adj(A)

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📌 Condition:

• Determinant ≠ 0

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❗ 9. Singular and Non-Singular Matrices

• Singular → $|A| = 0$∣A∣=0
• Non-Singular → $|A| \neq 0$∣A∣=0

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🧩 10. Solving Linear Equations (Cramer’s Rule)

For system:$$a_1x + b_1y + c_1z = d_1$$a1​x+b1​y+c1​z=d1​ $$a_2x + b_2y + c_2z = d_2$$a2​x+b2​y+c2​z=d2​ $$a_3x + b_3y + c_3z = d_3$$a3​x+b3​y+c3​z=d3​

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Solution:

$$x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta}$$x=ΔΔx​​,y=ΔΔy​​,z=ΔΔz​​

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$x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta}$x=ΔΔx​​,y=ΔΔy​​,z=ΔΔz​​

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📊 11. Applications of Determinants

• Solving equations
• Finding inverse
• Geometry (area, collinearity)
• Engineering & physics
• Computer graphics

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📌 12. Important Results

1. Determinant of Identity Matrix

= 1

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2. Determinant of Triangular Matrix

= Product of diagonal elements

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3. Determinant of Transpose

$$|A^T| = |A|$$∣AT∣=∣A∣

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❗ Common Mistakes

• Sign errors in expansion
• Wrong cofactor calculation
• Forgetting determinant condition for inverse
• Mistakes in row operations

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🧠 Exam Tips

• Practice 3×3 expansion
• Learn properties thoroughly
• Use shortcuts
• Solve NCERT examples

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📚 Practice Questions

1. Evaluate determinant
2. Find minors & cofactors
3. Check collinearity
4. Solve equations using Cramer’s rule

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🎯 Conclusion

Determinants simplify complex algebraic problems and are essential for solving systems of equations. Understanding properties, expansion methods, and applications will help you score high in board exams and competitive tests.