Matrices: Definition and Basic Concepts
A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns.
NoteIn the context of mathematics, matrices are fundamental tools used in various fields, including linear algebra, computer graphics, and data analysis.
Elements, Rows, Columns, and Order
- A matrix is typically denoted by a capital letter, such as A, B, or M.
- The individual entries in a matrix are called elements.
- The horizontal lines of elements are called rows, while the vertical lines are called columns.
- The order (or dimension) of a matrix is described by the number of rows and columns it contains.
An $m × n$ matrix has $m$ rows and $n$ columns.
NoteThe order of a matrix is always written as "rows × columns". A 2 × 3 matrix has 2 rows and 3 columns, not the other way around.
ExampleConsider the following matrix \( A \): \[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \] This is a \(3 \times 3\) matrix (3 rows and 3 columns). The element in the second row and third column is 6.
Matrix Algebra
Equality of Matrices
Two matrices are considered equal if they have the same order and all corresponding elements are equal.
ExampleIf \[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \] and \[ B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \] then \[ A = B. \]
Addition and Subtraction
Matrices of the same order can be added or subtracted element by element.
ExampleIf \[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \] and \[ B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}, \] then: \[ A + B = \begin{bmatrix} 1 + 5 & 2 + 6 \\ 3 + 7 & 4 + 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} \]\[ A - B = \begin{bmatrix} 1 - 5 & 2 - 6 \\ 3 - 7 & 4 - 8 \end{bmatrix} = \begin{bmatrix} -4 & -4 \\ -4 & -4 \end{bmatrix} \]
Scalar Multiplication
A matrix can be multiplied by a scalar (a single number) by multiplying each element of the matrix by that scalar.
ExampleIf \[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \] and $k = 3$, then: \[ 3A = \begin{bmatrix} 3(1) & 3(2) \\ 3(3) & 3(4) \end{bmatrix} = \begin{bmatrix} 3 & 6 \\ 9 & 12 \end{bmatrix} \]
Matrix Multiplication
- Matrix multiplication is more complex than addition or scalar multiplication.
- To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.
For matrices $A (m × n)$ and $B (n × p)$, their product $AB$ is an $m × p$ matrix.
ExampleIf \[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \] Then, \[ AB = \begin{bmatrix} (1 \times 5) + (2 \times 7) & (1 \times 6) + (2 \times 8) \\ (3 \times 5) + (4 \times 7) & (3 \times 6) + (4 \times 8) \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} \]
Common MistakeMatrix multiplication is not commutative. In general, AB ≠ BA, even when both products are defined.
Properties of Matrix Multiplication
- Associativity: (AB)C = A(BC)
- Distributivity: A(B + C) = AB + AC and (A + B)C = AC + BC
- Non-commutativity: AB ≠ BA (in general)
Special Matrices
Identity Matrix
The identity matrix, denoted as $I$, is a square matrix with 1s on the main diagonal and 0s elsewhere. When multiplied with any matrix $A$ of compatible size, $AI = IA = A$.
ExampleThe $3 \times 3$ identity matrix is: \[ I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]
Zero Matrix
The zero matrix, often denoted as 0, is a matrix where all elements are zero. Adding a zero matrix to any matrix $A$ results in $A$.
Determinants and Inverses
Determinants
- The determinant is a scalar value that can be computed from a square matrix.
- It provides important information about the matrix and is crucial in finding the inverse of a matrix.
For a 2 × 2 matrix A = $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is:
$$\det(A) = ad - bc$$
ExampleFor \[ A = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}, \] the determinant is: \[\det(A) = (3 \times 4) - (2 \times 1) = 12 - 2 = 10. \]
For larger matrices, determinants can be calculated using technology or more advanced techniques.
Determinant of a $3\times 3$ Matrix
The determinant of a \(3 \times 3\) matrix provides valuable information about the matrix, including whether it is invertible and the volume scaling factor of a transformation.
Given a matrix: \[ A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \]
its determinant is calculated using cofactor expansion along the first row:
\[\det(A) = a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \begin{vmatrix} d & e \\ g & h \end{vmatrix} \]
Expanding further using the determinant of a \(2 \times 2\) matrix: \[\det(A) = a (ei - fh) - b (di - fg) + c (dh - eg) \]
NoteThis determinant is essential in various applications, such as solving linear systems using Cramer's rule, computing matrix inverses, and analyzing geometric transformations.
Inverse Matrices
The inverse of a square matrix A, denoted A⁻¹, is a matrix that, when multiplied with A, gives the identity matrix: AA⁻¹ = A⁻¹A = I.
For a 2 × 2 matrix A = $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, if det(A) ≠ 0, the inverse is:
$$ A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $$
ExampleFor the matrix: \[ A = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} \] The determinant is calculated as: \[\det(A) = (3 \times 4) - (2 \times 1) = 12 - 2 = 10 \] The inverse of $A$ is given by: \[ A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} 4 & -2 \\ -1 & 3 \end{bmatrix} = \begin{bmatrix} 0.4 & -0.2 \\ -0.1 & 0.3 \end{bmatrix} \]
NoteNot all matrices have inverses. A matrix must be square and have a non-zero determinant to have an inverse.
Applications of Matrices
Solving Systems of Linear Equations
- Matrices can be used to solve systems of linear equations efficiently.
- A system of linear equations can be written in the form $Ax = b$, where $A$ is the coefficient matrix, $x$ is the vector of variables, and $b$ is the vector of constants.
Consider the system of equations: \[ x + 2y = 5 \] \[ 3x - y = 2 \] This can be written as: \[ \begin{bmatrix} 1 & 2 \\ 3 & -1 \end{bmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 2 \end{pmatrix} \] To solve this, we multiply both sides by $A^{-1}$: \[ \begin{pmatrix} x \\ y \end{pmatrix} = A^{-1} \begin{pmatrix} 5 \\ 2 \end{pmatrix} \]
Coding and Decoding Messages
- Matrices can be used in cryptography for encoding and decoding messages.
- A simple method involves converting letters to numbers, arranging them in a matrix, and then multiplying by an encoding matrix.
Suppose we want to encode the message "HELLO" using the encoding matrix: \[ A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix} \] First, convert letters to numbers (A=0, B=1, ..., Z=25): $H=7$, $E=4$, $L=11$, $L=11$, $O=14$. Arrange these in a matrix: \[\begin{bmatrix} 7 & 11 \\ 4 & 11 \end{bmatrix} \] Multiply by the encoding matrix: \[\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} 7 & 11 \\ 4 & 11 \end{bmatrix} = \begin{bmatrix} 18 & 33 \\ 19 & 44 \end{bmatrix} \] The encoded message is now represented by these numbers.
Modeling Real-Life Problems
- Matrices can model various real-world scenarios, such as population growth, economic systems, and computer graphics transformations.
In a simple predator-prey model, if $x$ represents the prey population and $y$ the predator population, their change over time can be modeled as: \[\begin{pmatrix} \Delta x \\\Delta y \end{pmatrix} = \begin{bmatrix} a & -b \\ c & -d \end{bmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \] Where $a, b, c,$ and $d$ are constants representing birth and death rates.
TipWhen solving real-world problems using matrices, always interpret your results in the context of the original problem. Mathematical solutions sometimes need to be adjusted to make sense in the real world.
