Overview
This lecture covers the basics of matrix addition and subtraction, including requirements for these operations, worked examples, and key properties of matrix addition.
Matrix Addition and Subtraction
- To add or subtract matrices, they must have the same dimensions.
- The result of addition or subtraction has the same size as the original matrices.
- Addition is performed by adding corresponding entries from each matrix.
- Subtraction is performed by subtracting corresponding entries in order (A - B ≠ B - A).
- Matrix addition or subtraction is not possible if matrices have different sizes.
Example Problems
- D - C: Both are 2×3 matrices; subtract corresponding entries to get the resulting 2×3 matrix.
- C - B: Not possible; C is 2×3, B is 2×2 (different sizes).
- A + B: Both are 2×2 matrices (with variables possible); add corresponding entries for the 2×2 result.
Using a Calculator for Matrix Operations
- Numeric matrices can be entered into a calculator using the matrix menu.
- Enter matrices under ‘Edit,’ specify size, input values, then perform operations using the ‘Names’ menu.
- Matrix operations with variables cannot be done on a calculator and must be performed by hand.
Properties of Matrix Addition
- Commutative Property: A + B = B + A (only for addition, not subtraction).
- Associative Property: (A + B) + C = A + (B + C), as long as matrices are the same size.
- Identity Property: Adding a zero matrix (all zeros) of the same size does not change the matrix (A + 0 = A).
- Inverse Property: Each matrix has an additive inverse (all entries have opposite signs); adding a matrix to its inverse gives the zero matrix.
Key Terms & Definitions
- Matrix — a rectangular array of numbers arranged in rows and columns.
- Dimensions — the number of rows and columns in a matrix (e.g., 2×3).
- Zero Matrix (Additive Identity) — a matrix with all entries equal to zero.
- Additive Inverse — a matrix where each entry is the opposite sign of the original.
- Commutative Property — order of addition does not affect the result.
- Associative Property — grouping of addition does not affect the result.
Action Items / Next Steps
- Review and practice addition and subtraction of matrices with same and different sizes.
- Complete assigned problems from pages 2–4 of your lecture notes.