Lecture on Determinants

Introduction

• GodGlasses channel: mess and fun learning.
• Today's topic: one-shot on determinants.
• Importance: Combined with matrices, worth 10 marks.
• Resources available: question bank and sample papers on the God Classes app.

What is a Determinant?

• A determinant is a real number associated with a square matrix.
• Notation: Can be represented as |A|.
• Calculation examples:
  • 2x2 Matrix: For matrix [\begin{bmatrix} 1 & 3 \ 2 & 5 \end{bmatrix}], determinant = ((15) - (32) = -1).
  • 3x3 Matrix: Process involves eliminating rows and columns to calculate smaller 2x2 determinants.

Minor and Cofactor

• Minor: Determinant of a submatrix formed by deleting a row and column.
  • Example: For [\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}], minor of element 1 is 4.
• Cofactor: Minor adjusted by a sign based on position in matrix.
  • Example: Change sign if sum of row and column indices is odd.

Properties of Determinants

1. Determinant is 0 if all elements in a row/column are 0.
2. Determinant equals product of diagonal elements if non-diagonal elements are 0 on one side.
3. Transposing a matrix does not change its determinant.
4. Interchanging rows/columns changes the sign of determinant.
5. Identical or proportional rows/columns result in determinant of 0.
6. Multiplying a row/column by constant k, multiplies determinant by k.
7. Product of determinants equals determinant of product (Det(AB) = Det(A)Det(B)).
8. Expanding determinant by row/column cofactor sum results in original determinant.

Solving Equations using Determinants

• Cramer's Rule: Solve linear equations using determinants if system is consistent (D ≠ 0).
• Matrix Inverse Method: Solve for variables using A inverse if determinant is non-zero.

Applications

• Area of Triangle: Use determinant formula to calculate area using vertices.
• System of Equations: Solve systems using inverse matrices or Cramer's Rule.

Singular and Non-Singular Matrices

• Singular: Determinant is 0, inverse does not exist.
• Non-Singular: Determinant is non-zero, inverse exists.

Finding Inverses

• Formula: A inverse = adjoint(A) / determinant(A).
• Adjoint: Transpose of cofactor matrix.

Solution of Linear Equations by Matrix Method

• Check determinant: if zero, check consistency for solutions.
• Unique solution exists if determinant is non-zero.

Practice Problems

• Calculating determinants for given matrices.
• Solving systems using inverse and Cramer's Rule.

Conclusion

• Review determinants thoroughly for board exams.
• Utilize available resources for additional practice.