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.
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.