### Notes from Toppers

**Determinants: JEE Toppers’ Detailed Notes**

**1. Concept of a Determinant**

- A determinant is a scalar value associated with a square matrix.
- It can be calculated by using the Laplace expansion or by using the row or column operations.
- The determinant of a matrix is denoted by det(A) or |A|.

**(Reference: NCERT Class 12, Chapter 4, Determinants)**

**2. Properties of Determinants**

- The determinant of a triangular matrix is the product of its diagonal elements.
- The determinant of a diagonal matrix is the product of its diagonal elements.
- The determinant of a symmetric matrix is always non-negative.
- If two rows or columns of a matrix are interchanged, the determinant changes sign.
- If a row or column of a matrix is multiplied by a constant, the determinant is multiplied by that constant.

**(Reference: NCERT Class 12, Chapter 4, Determinants)**

**3. Minors and Cofactors**

- A minor of a matrix is the determinant of a submatrix obtained by deleting a row and a column of the original matrix.
- A cofactor of a matrix is the minor of the matrix multiplied by (-1) raised to the power of the sum of the row and column indices of the deleted element.

**(Reference: NCERT Class 12, Chapter 4, Determinants)**

**4. Cramer’s Rule**

- Cramer’s rule is a method for solving systems of linear equations with the same number of equations and variables.
- It involves calculating the determinant of the coefficient matrix and the determinants of the matrices obtained by replacing each column of the coefficient matrix with the column of constants.

**(Reference: NCERT Class 12, Chapter 4, Determinants)**

**5. Applications of Determinants**

- Determinants are used to find the area of polygons.
- Determinants are used to find the volume of parallelepipeds.
- Determinants are used to check the singularity of matrices.
- Determinants are used in linear algebra to solve systems of linear equations, find matrix inverses, and calculate eigenvalues and eigenvectors.

**(Reference: NCERT Class 12, Chapter 4, Determinants)**

**6. Determinants of Special Matrices**

- The determinant of a triangular matrix is the product of its diagonal elements.
- The determinant of a diagonal matrix is the product of its diagonal elements.
- The determinant of a symmetric matrix is always non-negative.

**(Reference: NCERT Class 12, Chapter 4, Determinants)**

**7. Eigenvalues and Eigenvectors**

- An eigenvalue of a matrix is a scalar that, when multiplied by the corresponding eigenvector, produces another vector that is parallel to the eigenvector.
- The determinant of a matrix is equal to the product of its eigenvalues.

**(Reference: NCERT Class 12, Chapter 4, Determinants)**

**8. Applications of Determinants in Geometry**

- Determinants are used to find the area of a parallelogram.
- Determinants are used to find the volume of a tetrahedron.
- Determinants are used to find the equation of a plane in three-dimensional space.

**(Reference: NCERT Class 12, Chapter 4, Determinants)**

**9. Applications in Linear Algebra**

- Determinants are used to solve systems of linear equations.
- Determinants are used to find matrix inverses.
- Determinants are used to calculate eigenvalues and eigenvectors.

**(Reference: NCERT Class 12, Chapter 4, Determinants)**

**10. Solving JEE-level Problems**

- JEE-level problems on determinants can be solved by using the concepts and properties of determinants discussed above.
- It is important to practice solving a variety of JEE-level problems in order to develop problem-solving skills and reinforce understanding of the concepts.

**(Reference: JEE Main and Advanced Previous Years’ Question Papers)**