Overview
This lecture covers how to multiply matrices by scalars (real numbers), and demonstrates combining scalar multiplication with matrix addition using worked examples.
Scalar Multiplication of Matrices
- Scalar multiplication means multiplying every entry in a matrix by a real number (scalar).
- To multiply a matrix by a scalar k, distribute k to each entry in the matrix.
- Example: To find (-1/2) × matrix C, multiply every entry of C by -1/2.
- Maintain the sign of each entry when multiplying by the scalar.
- Perform the arithmetic for each entry for the final result.
Example: Multiplying Matrix by Scalar
- Example: Multiply -1/2 by matrix C:
- Apply -1/2 to each entry.
- Results for each entry: -3/2, -3, 9/2, -5/2, -7/2, 1/2.
Combining Scalar Multiplication and Matrix Addition
- To combine expressions like -6B + 7A, matrices B and A must be the same size.
- Distribute -6 to each entry of matrix B, and 7 to each entry of matrix A.
- Simplify each resulting matrix before adding.
- Add corresponding entries to form the final matrix.
Example: -6B + 7A
- Multiply -6 to B’s entries: -6, 66, -18y, -108.
- Multiply 7 to A’s entries: -14, 28x, 7y, 56.
- Add corresponding entries: -20, 28x+66, -11y, -52.
Key Terms & Definitions
- Matrix — A rectangular array of numbers arranged in rows and columns.
- Scalar — A real number used to multiply a matrix.
- Scalar Multiplication — Multiplying every entry in a matrix by the same real number.
- Corresponding Entries — Entries in the same position in two matrices of the same size.
Action Items / Next Steps
- Practice scalar multiplication and addition of matrices using provided textbook exercises.