Matrices and Scalar Multiplication

Key Concepts

• Scalar: A real number used to multiply each entry of a matrix.
• Matrix: A rectangular array of numbers arranged in rows and columns.

Multiplying a Matrix by a Scalar

• Process involves multiplying each entry of the matrix by a real number (scalar).
• Example: Given a matrix C and scalar -1/2, distribute the scalar to each element of C.
  • Maintain positive and negative signs during multiplication.

Example 1: -1/2 * Matrix C*

• Multiply -1/2 with each entry in matrix C.
• Resulting entries:
  • Negative 3/2
  • Negative 3
  • Positive 9/2
  • Negative 5/2
  • Negative 7/2
  • Positive 1/2

Operations Involving Multiple Matrices

• Discusses the operation of -6B + 7A.
• Requirement: Matrices A and B must be of the same size for addition.
  • Both are 2x2 matrices in this example.

Example 2: -6B + 7A

• Step 1: Distribute scalars to corresponding matrices.

  • Multiply each entry of matrix B by -6.
  • Multiply each entry of matrix A by 7.

• Step 2: Simplify each entry of the matrices before adding.

  • Matrix B entries:
    • Negative 6, 66, -18y, -108
  • Matrix A entries:
    • Negative 14, 28x, 7y, 56

• Step 3: Add corresponding entries to get the resulting matrix.

  • Resulting matrix entries:
    • Negative 20
    • 28x + 66
    • Negative 11y
    • Negative 52

• Note: Parentheses can be used around the sum entries for clarity.

Conclusion

• Scalar multiplication involves straightforward arithmetic applied to each matrix entry.
• Ensure matrices are of the same size when performing operations involving multiple matrices.