Lecture on Matrices

Definition of a Matrix

• A matrix is an array of numbers organized into rows and columns.

Order of a Matrix

• Order: Described by the number of rows x number of columns (e.g., 2x3).
• Example: Matrix A with elements 2, 7, -4, 6, 3, 5 has an order of 2x3.
• Rows are horizontal; columns are vertical.

Identifying Elements in a Matrix

• Elements are denoted as Aij where i is the row number and j is the column number.
• Example: Element A23 is in the second row, third column.
• Practice:
  • A12 = 7 (1st row, 2nd column)
  • A21 = 6 (2nd row, 1st column)

Example with Matrix B

• Contains elements: 4, 3, 7, -2, 5, 6, -4, 9, -3, 8, 1, -7.
• Order: 3x4 (3 rows, 4 columns).
• Elements:
  • B11 = 4 (1st row, 1st column)
  • B23 = -4 (2nd row, 3rd column)
  • B14 = -2 (1st row, 4th column)
  • B34 = -7 (3rd row, 4th column)

Identifying Matrices Order

• Matrix C (3, -7, 8, -5, 2, -1): Order is 2x3.
• Matrix D (4, 5, -2, 7, 3, -6): Order is 3x2.
• Matrix E (8): Order is 1x1 (single element).
• Matrix F (7, 4, -5, 11): Order is 1x4.
• Matrix G (3, 1, 7, 2, 6, -4, 9, 0, 3): Order is 3x3 (square matrix).
• Matrix H (2, 1, 7, -3, 6, -2, 5, 4): Order is 2x4.
• Square Matrices: C (2x2), G (3x3).

Matrix Operations

Adding Matrices

• Can only add matrices of the same order.
• Add corresponding elements:
  • Example: A = [2, 3, 5, -4], B = [7, 4, -3, 5]
  • Sum = [9, 7, 2, 1]

Multiplying a Matrix by a Scalar

• Multiply each element by the scalar.
• Example: 4 x A = [8, 12, 20, -16]

Subtracting Matrices

• Subtract corresponding elements:
  • A - B = [-5, -1, 8, -9]

Conclusion

• Understanding matrices involves knowing their order, identifying elements, and performing basic operations like addition, subtraction, and scalar multiplication.
• Further resources available in video description.