Overview
This lecture explains how to multiply two matrices, determine if multiplication is possible, and find each element of the resulting product matrix step by step.
Determining Matrix Multiplication Possibility
- Matrix A has order 2x3 (2 rows, 3 columns).
- Matrix B has order 3x2 (3 rows, 2 columns).
- Matrix multiplication is possible if columns of A = rows of B (here, both are 3).
- The resulting matrix AB has order 2x2 (rows of A, columns of B).
Setting Up the Product Matrix
- The product matrix AB will have 2 rows and 2 columns.
- Each cell is identified by (row, column), e.g., cell (1,1) is row 1, column 1.
Calculating Elements of AB
- Cell (1,1):
- Multiply row 1 of A by column 1 of B: (1×3) + (2×2) + (-3×-1) = 3 + 4 + 3 = 10.
- Cell (1,2):
- Multiply row 1 of A by column 2 of B: (1×1) + (2×4) + (-3×5) = 1 + 8 - 15 = -6.
- Cell (2,1):
- Multiply row 2 of A by column 1 of B: (4×3) + (0×2) + (-2×-1) = 12 + 0 + 2 = 14.
- Cell (2,2):
- Multiply row 2 of A by column 2 of B: (4×1) + (0×4) + (-2×5) = 4 + 0 - 10 = -6.
Resulting Matrix
- The product matrix AB is:
Key Terms & Definitions
- Order (of a matrix) — Number of rows × number of columns.
- Matrix multiplication — Combining two matrices by multiplying corresponding elements and adding results to fill each cell in the product matrix.
- Cell (i, j) — The element located at row i, column j of a matrix.
Action Items / Next Steps
- Practice multiplying matrices of varying sizes.
- Review properties and conditions required for matrix multiplication.