Class Notes on Matrices
Summary
In this class, we focused on understanding the concept of matrices including how to determine their order, how to identify specific elements within them, and basic operations like addition, subtraction, and scalar multiplication.
Definition of a Matrix
- A matrix is an array of numbers organized into rows and columns.
Order of a Matrix
- The order of a matrix is described as "rows by columns."
- Example:
- Matrix A: ( \begin{bmatrix} 2 & 7 & -4 \ 6 & 3 & 5 \end{bmatrix} ) - Order: 2 by 3 (two rows and three columns)
Identifying Elements in a Matrix
- Elements in a matrix are identified using two indices, the first representing the row and the second the column.
- Example:
- In Matrix A ( \begin{bmatrix} 2 & 7 & -4 \ 6 & 3 & 5 \end{bmatrix} ):
- Element ( A_{23} ) (Second row, third column) is 5.
- Element ( A_{12} ) is 7.
- Element ( A_{21} ) is 6.
Matrix B
- Matrix B: ( \begin{bmatrix} 4 & 3 & 7 & -2 \ 5 & 6 & -4 & 9 \ -3 & 8 & 1 & -7 \end{bmatrix} )
- Order: 3 by 4
- Identified Elements:
- ( B_{11} ) = 4
- ( B_{23} ) = -4
- ( B_{14} ) = -2
- ( B_{34} ) = -7
Determining the Order of Other Matrices
- Matrix C: ( \begin{bmatrix} 3 & -5 \ 2 & -1 \end{bmatrix} ) - 2 by 2, square matrix
- Matrix D: ( \begin{bmatrix} 4 & 5 \ -2 & 7 \ 3 & -6 \end{bmatrix} ) - 3 by 2
- Matrix E: ( \begin{bmatrix} 8 \end{bmatrix} ) - 1 by 1, square matrix
- Matrix F: ( \begin{bmatrix} 7 & 4 & -5 & 11 \end{bmatrix} ) - 1 by 4
- Matrix G: ( \begin{bmatrix} 3 & 1 & 7 \ 2 & 6 & -4 \ 9 & 0 & 3 \end{bmatrix} ) - 3 by 3, square matrix
- Matrix H: ( \begin{bmatrix} 2 & 1 & 7 & -3 \ 6 & -2 & 5 & 4 \end{bmatrix} ) - 2 by 4
Basic Operations
Addition of Matrices
- Only matrices of the same order can be added.
- Example with Matrix A and Matrix B:
- Result: ( \begin{bmatrix} 9 & 7 & 2 \ 1 & 1 & 1 \end{bmatrix} )
Scalar Multiplication
- Multiplying every element of the matrix by the scalar.
- Example with Matrix A multiplied by 4:
- Result: ( \begin{bmatrix} 8 & 12 & 20 \ -16 & -12 & 20 \end{bmatrix} )
Subtraction of Matrices
- Subtract corresponding elements of two matrices of the same order.
- Example with Matrix A and Matrix B:
- Result: ( \begin{bmatrix} -5 & -1 & 8 \ -9 & -9 & 8 \end{bmatrix} )
Conclusion
- These basic operations and properties form the foundation of working with matrices which are crucial for various applications in mathematics and science.
Additional Resources
- For further reading and practice, check the description section of the video for links to more pre-calculus resources.
This lecture equipped you with the foundational knowledge to handle matrices and their basic operations which will be applicable in various mathematical problems and contexts.