Adding and Subtracting Matrices

Learning Outcomes

  • Use matrices to solve real-world problems (e.g., soccer team statistics).

Definition and Basics of Matrices

  • Matrix: A rectangular array of numbers defined by rows and columns.

    • Row: A horizontal set of entries.

    • Column: A vertical set of entries.

    • Element/Entry: A single number within the matrix.

Matrix Notation

  • Matrices are typically enclosed in square brackets [ ] or

    parentheses ( ).

  • Often denoted with capital letters (A, B, C, …).

Dimensions of a Matrix

  • Defined as m x n, where:

    • m = number of rows

    • n = number of columns

  • Example: A matrix with dimensions 3 x 3 has 3 rows and 3 columns.

Understanding Matrix Entries

  • Each entry is identified by its position, typically written as aija_{ij}, where:

    • i = row number

    • j = column number

  • Pronunciation of subscripts:

    • Example: a13a_{13} is pronounced "a one three."

Types of Matrices

  • Square Matrix: Dimensions are n x n (equal rows and columns).

  • Row Matrix: A matrix with only one row (1 x n).

  • Column Matrix: A matrix with only one column (m x 1).

Matrix Operations

Addition of Matrices

  • Can only be performed when two matrices are of the same dimensions (e.g., 3 x 3 + 3 x 3).

  • When adding matrices A and B:

    • C=A+BC = A + B

    • cij=aij+bijc{ij} = a{ij} + b_{ij}

  • Commutative Property: A+B=B+AA + B = B + A

  • Associative Property: (A+B)+C=A+(B+C)(A + B) + C = A + (B + C)

Example of Addition

  1. Given:
    A=[4amp;13amp;2] and B=[5amp;90amp;7]A = \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix} \text{ and } B = \begin{bmatrix} 5 & 9 \\ 0 & 7 \end{bmatrix}

  2. Solution:
    A+B=[4+5amp;1+93+0amp;2+7]=[9amp;103amp;9]A + B = \begin{bmatrix} 4+5 & 1+9 \\ 3+0 & 2+7 \end{bmatrix} = \begin{bmatrix} 9 & 10 \\ 3 & 9 \end{bmatrix}

  • Similar to addition, can only be performed on matrices of the same dimensions.

  • When subtracting matrices A and B:

    • D=A−BD = A - B

    • dij=aij−bijd{ij} = a{ij} - b_{ij}

Example of Subtraction

  1. Given:
    A=[2amp;30amp;1],B=[8amp;15amp;4]A=\begin{bmatrix}2 & 3\\ 0 & 1\end{bmatrix},B=\begin{bmatrix}8 & 1\\ 5 & 4\end{bmatrix}

  2. Solution:
    A−B=[2−8amp;3−10−5amp;1−4]=[−6amp;2−5amp;−3]A - B = \begin{bmatrix} 2-8 & 3-1 \\ 0-5 & 1-4 \end{bmatrix} = \begin{bmatrix} -6 & 2 \\ -5 & -3 \end{bmatrix}

    Tip for Success

  • Handle negative numbers carefully during matrix subtraction to ensure accuracy.

  • Always remember: Row comes first, then column when locating entries.