Notes on Matrices

Definition of a Matrix

• A matrix is an array of numbers arranged in rows and columns.

Order of a Matrix

• The order of a matrix is defined by its number of rows and columns.
• Example: For matrix A with elements 2, 7, -4, 6, 3, 5:
  • Order: 2 rows and 3 columns → 2 x 3 matrix

Identifying Elements in a Matrix

• Elements are identified by their row and column position.
• Notation: Element a_{ij} where i is the row and j is the column.
• Example for matrix A:
  • Element a_{23} = 5 (2nd row, 3rd column)
  • Element a_{12} = 7 (1st row, 2nd column)
  • Element a_{21} = 6 (2nd row, 1st column)

Another Example: Matrix B

• Matrix B has elements: 4, 3, 7, -2, 5, 6, -4, 9, -3, 8, 1, -7.
• Order: 3 rows and 4 columns → 3 x 4 matrix
• Example of identifying elements in matrix B:
  • Element b_{11} = 4
  • Element b_{23} = -4
  • Element b_{14} = -2
  • Element b_{34} = -7

Identifying the Order of Multiple Matrices

• Matrix C: 2 x 2 (square matrix)
• Matrix D: 3 x 2
• Matrix E: 1 x 1 (single number)
• Matrix F: 1 x 4
• Matrix G: 3 x 3 (square matrix)
• Matrix H: 2 x 4

Adding Matrices

• Matrices can only be added if they have the same order.
• Example: Add matrix A (2, 3, 5, -4) and matrix B (7, 4, -3, 5):
  • Sum:
    • (2+7, 3+4, 5+(-3), -4+5)
    • Result: (9, 7, 2, 1)

Multiplying a Matrix by a Scalar

• To multiply a matrix by a scalar, multiply every element by that scalar.
• Example: 4A for matrix A (2, 3, 5, -4):
  • Result: (8, 12, 20, -16)

Subtracting Matrices

• Matrices can be subtracted in the same way as they are added, ensuring they have the same order.
• Example: Subtract matrix A from matrix B:
  • Result:
    • (2 - 7, 3 - 4, 5 - (-3), -4 - 5)
    • Result: (-5, -1, 8, -9)

Conclusion

• Understanding the order of matrices and how to identify elements is essential.
• Addition and subtraction of matrices require matching dimensions.

Additional Resources

• For more videos on pre-calculus, check the description section for links.