Lecture on Matrices: Basic Operations

Key Topics Covered

• Addition and Subtraction of Matrices
• Properties of Matrix Addition

Matrix Operations

Addition

• Definition: Two matrices can be added only if they have the same dimensions.
• Resulting Matrix: The resulting matrix will have the same dimensions as the original matrices.
• Method: Add corresponding entries from each matrix.

Subtraction

• Definition: Similar to addition, matrices must be the same size.
• Resulting Matrix: The resulting matrix will have the same dimensions as the original matrices.
• Order: Important to maintain order; A - B ≠ B - A.
• Method: Subtract corresponding entries.

Examples

Example 1: Subtracting Matrices D and C

• Matrices Provided: D and C, both are 2x3 matrices.
• Calculation:
  • Subtract entries of C from D:
  • Resulting matrix:
    • 1st row: [6-3, -2-6, 12-(-9)] => [3, -8, 21]
    • 2nd row: [-8-5, 3-7, 4-(-1)] => [-13, -4, 5]
• Final Result: 2x3 matrix:
  • [ [3, -8, 21], [-13, -4, 5] ]

Example 2: Using Calculator for Matrix Operations

• Steps:
  • Enter matrices using matrix menu on calculator.
  • Perform subtraction D - C using calculator functions.

Example 3: C minus B

• Matrices:
  • C (2x3), B (2x2)
• Outcome: Operation not possible as dimensions don't match.

Example 4: A plus B

• Matrices:
  • A (2x2), B (2x2)
• Calculation:
  • Add corresponding entries:
    • Resulting matrix:
      • [ [-2+1, 4x-11], [y+3y, 8+18] ]
      • Simplifies to [ [-1, 4x-11], [4y, 26] ]

Properties of Matrix Addition

Commutative Property

• The order of addition does not matter as long as matrices are the same size.
• A + B = B + A

Associative Property

• The grouping of matrices doesn't change the result.
• (A + B) + C = A + (B + C)

Identity Property

• Adding a zero matrix (all entries are zeros) to any matrix yields the original matrix.

Inverse Property

• The inverse of a matrix is obtained by changing the sign of each entry.
• Adding a matrix and its inverse results in a zero matrix.

Examples of Properties

Example 5: Additive Inverse

• Matrix B:
  • Change sign of each entry.
  • Result: [ [-1, 11], [-3y, -18] ]

Example 6: Additive Identity

• Matrix D (2x3):
  • Identity matrix is a zero matrix of the same size (2x3).