Lecture Notes on Matrices

Introduction

• New session on Matrices.
• Presented in English for students who do not understand Hindi.

What is a Matrix?

• A matrix is a rectangular arrangement of numbers or symbols.
• Types of numbers in matrices:
  • Real Numbers: Forms a real matrix.
  • Complex Numbers: Forms a complex matrix.

Structure of a Matrix

• Rows: Horizontal arrangement.
• Columns: Vertical arrangement.
• Example:
  • Matrix: 1 2 3 4 5 6
  • Rows: (1, 2, 3) - First Row; (4, 5, 6) - Second Row.
  • Columns: (1, 4) - First Column; (2, 5) - Second Column; (3, 6) - Third Column.

Order of a Matrix

• Represented as m x n (m by n).
• Example: For matrix: 1 2 3 4 5 6
  • Order: 2 rows x 3 columns = 2 x 3.

Types of Matrices

1. Row Matrix: Only one row (1 x n).

   • Example: (1, 2, 3, 4, 5) - Order 1 x 5.

2. Column Matrix: Only one column (m x 1).

   • Example: (1, 2, 3) - Order 3 x 1.

3. Zero Matrix: All entries are zero (null matrix).

   • Example: 2 x 2 zero matrix:
   0 0 0 0

4. Square Matrix: Number of rows equals number of columns (m x m).

   • Example: 2 x 2 matrix:
   1 2 3 4

5. Diagonal Matrix: Square matrix where non-diagonal elements are zero.

   • Example:
   1 0 0 0 2 0 0 0 3

6. Scalar Matrix: Diagonal elements are the same.

   • Example:
   2 0 0 0 2 0 0 0 2

7. Unit Matrix (Identity Matrix): Diagonal elements are 1.

   • Example:
   1 0 0 0 1 0 0 0 1

8. Upper Triangular Matrix: Non-zero elements above the diagonal.

   • Example:
   1 2 3 0 4 5 0 0 6

9. Lower Triangular Matrix: Non-zero elements below the diagonal.

   • Example:
   1 0 0 2 3 0 4 5 6

10. Sub Matrix: A matrix obtained from another by deleting rows and columns.

    • Example: Removing one row and one column.

11. Principal Sub Matrix: Sub matrix that is also square.

Equality of Matrices

• Two matrices are equal if all corresponding elements are equal.
  • Example: Matrix A = a b c d Compare with Matrix B: e f g h
• A = B if a = e, b = f, c = g, d = h.

Addition of Matrices

• Can only add matrices of the same order.
• Example: A = 1 2 3 4 B = 1 2 5 7 A + B = 2 4 8 11

Multiplication of Matrices

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• Example: A = 1 2 3 4 B = 1 2 3 4 5 6
• Resulting matrix order is m x p.

Scalar Multiplication

• Multiply every element by a constant (scalar).

Trace of a Matrix

• Sum of diagonal elements in square matrices.
• Denoted as tr(A).

Transpose of a Matrix

• Interchange rows and columns.
• Denoted as A^T or A'.
• Symmetric Matrix: A = A^T.
• Skew-Symmetric Matrix: A^T = -A (diagonal elements are 0).

Conclusion

• Overview of concepts covered in matrices including definitions, types, operations, and properties.
• Questions or doubts can be addressed in future sessions.