Matrix Algebra and Operations Summary

Matrices and Their Algebra

Introduction to Matrices

• A matrix is a rectangular array of numbers or expressions. It can be represented as:

  [
  A = \begin{bmatrix}
  a{11} & a{12} & … & a{1n} \ a{21} & a{22} & … & a{2n} \
  … & … & … & … \
  a{m1} & a{m2} & … & a_{mn}
  \end{bmatrix}
  ]

• Matrices can be viewed as a list of rows or columns.

Applications of Matrices

• Used in various fields such as:
• Solving systems of linear equations
• Operations research and linear programming
• Computer graphics
• Engineering applications
• Statistical analysis

Goals of Matrix Study

• Learn how to utilize matrix algebra to express systems of equations as matrix equations
• Find the inverse of a matrix to solve linear equations
• Understand the relationship between matrix inverses and solutions of linear systems

Matrix Size and Order

• The order of a matrix is indicated by its dimensions (m \times n), where (m) is the number of rows and (n) is the number of columns.

  • Example:

    [
    \begin{bmatrix}
    1 & 2 & 3 \
    4 & 5 & 6
    \end{bmatrix}
    z
    ]

  • This matrix has an order of (2 \times 3).

• Exercises: Determine the order of various matrices.

Matrix Notation and Terminology

• Capital letters (A, B) are used for matrices; lower-case bold letters ((\mathbf{x})) are vectors.
• The entry in the (i)-th row and (j)-th column is denoted (a_{ij}).
• A vector can be either a column vector or a row vector, with column vectors being emphasized.
• A 1 (\times) 1 matrix corresponds to a scalar (real number).

Operations on Matrices

• Addition/Subtraction: Matrices can be added or subtracted if they are of the same order.
  • (A + B = C) where:
  • (c{ij} = a{ij} + b_{ij}).
• Scalar Multiplication: When a matrix is multiplied by a scalar, each entry is multiplied by that scalar.
  • Example: If (s) is a scalar and (A) is a matrix, then (sA = As).

Special Types of Matrices

• Square Matrix: Has the same number of rows and columns ((n \times n)).
  • Example:
    [
    \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}
    ]
• Diagonal Matrix: Non-zero entries exist only on the main diagonal.
• Identity Matrix: A diagonal matrix with ones on the main diagonal, denoted as (I_n).
• Block Matrix: Formed from smaller matrices combined into a larger matrix.

Matrix Addition and Subtraction

• Matrices can only be added/subtracted if they conform (same size).
• The zero matrix acts as the additive identity for matrices.
• Properties:
  • Commutative: (A + B = B + A)
  • Associative: (A + (B + C) = (A + B) + C)

Matrix Multiplication

• Two matrices can be multiplied if the number of columns in the first equals the number of rows in the second.
• The result's order will correspond to the outer dimensions.

Calculating Matrix Products

• The entry in the product ((AB){ij}) can be calculated using the sum of products from the row of the first matrix and the column of the second: [ (AB){ij} = \sum{k=1}^{n} a{ik}b_{kj}
  ]

Matrix Properties

• Associative: (A(BC) = (AB)C)
• Distributive:(A(B+C) = AB + AC)

Important Matrix Concepts

• Invertible Matrix: A matrix is invertible if there exists a matrix (C) such that (CA = I) and (AC = I).
  • If (A) is invertible, then (A^{-1}A = I).
• Homogeneous Systems: A system of equations that can always be expressed in the form (Ax = 0).
• Gaussian Elimination: A method to solve systems of equations through row-reduction to echelon forms.

Summary of Matrix Operations

• Operations with matrices include addition, subtraction, multiplication, and finding inverses.
• Understanding the relationships between matrix manipulations and linear equation solutions is crucial.
• Properties of matrix operations (associative, distributive, commutativity in addition) must also be understood for effective manipulation of matrix equations.