Applications and Properties of Matrices

Basic Definition: A scalar value that can be computed from the elements of a square matrix.
Non-Singular Matrix: A matrix is invertible if its determinant is non-zero.
Row Operations:
- Swapping two rows multiplies the determinant by -1.
- Multiplying a row by a scalar multiplies the determinant by that scalar.
- Adding a multiple of one row to another does not change the determinant.
Determinant of a Product: ( \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) ).
Triangular Matrices: The determinant is the product of the diagonal elements.

Definition: The matrix ( A^{-1} ) such that ( A \cdot A^{-1} = I ), where ( I ) is the identity matrix.
Existence: A matrix has an inverse only if it is square and its determinant is non-zero.
Calculation Using Determinants:
- For a 2x2 matrix ( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} ): [ A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} ]
Adjugate Method: For larger matrices, ( A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) ) where ( \text{adj}(A) ) is the adjugate of ( A ).

Application: Solves linear systems of equations using determinants.
Conditions: Applicable only for square matrices with a non-zero determinant.
Formula: For a system ( Ax = b ):
- Let ( x_i = \frac{\text{det}(A_i)}{\text{det}(A)} ), where ( A_i ) is matrix ( A ) with the i-th column replaced by vector ( b ).

Addition: ( C = A + B ) (element-wise addition).
Subtraction: ( C = A - B ) (element-wise subtraction).
Scalar Multiplication: ( B = kA ) where each element of ( A ) is multiplied by scalar ( k ).
Matrix Multiplication: ( C = A \times B ):
- Element ( c_{ij} = \sum_{k} a_{ik} \cdot b_{kj} ).
Transpose: Switches rows and columns of a matrix ( A^T ).
Associative and Distributive Properties:
- Associative: ( A(BC) = (AB)C ).
- Distributive: ( A(B + C) = AB + AC ).

Computer graphics utilize matrices for transformations, including scaling, rotation, and translation of images.
In statistics, matrices represent data sets and manage covariance matrices to examine relationships between variables.
Economically, input-output models employ matrices to analyze and illustrate complex economic systems.
Engineering relies on matrices for structural analysis and to model various systems encountered in design and construction.
Markov chains use transition matrices to represent probabilities across different states in stochastic processes.

The determinant is a unique scalar value derived from a square matrix's elements.
A matrix is classified as non-singular and therefore invertible if its determinant is not equal to zero.
Row operations alter the determinant:
- Swapping two rows results in a determinant multiplied by -1.
- Multiplying a row by a scalar changes the determinant by that scalar.
- Adding a multiple of one row to another retains the original determinant value.
For matrices A and B, the determinant of their product is equal to the product of their determinants: ( \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) ).
For triangular matrices, the determinant is calculated by multiplying the diagonal elements together.

The inverse of matrix A, denoted ( A^{-1} ), satisfies the equation ( A \cdot A^{-1} = I ), where I is the identity matrix.
A matrix can only possess an inverse if it is square and has a non-zero determinant.
For a 2x2 matrix represented as ( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} ), the inverse is given by: [ A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} ]
Larger matrices use the adjugate method for calculation: ( A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) ).

Cramer's Rule provides a technique for solving systems of linear equations through the use of determinants.
The rule applies exclusively to square matrices that have a non-zero determinant.
For any system represented as ( Ax = b ), the individual components of the solution can be found using:
- ( x_i = \frac{\text{det}(A_i)}{\text{det}(A)} ), where ( A_i ) is the matrix A altered by replacing the i-th column with vector b.

Addition of matrices is performed element-wise, yielding ( C = A + B ).
Subtraction is similar and calculated as ( C = A - B ) through element-wise operations.
Scalar multiplication involves multiplying each element of matrix A by a scalar ( k ), resulting in ( B = kA ).
Matrix multiplication, denoted by ( C = A \times B ), is executed using the formula:
- The entry at position ( c_{ij} ) is ( \sum_{k} a_{ik} \cdot b_{kj} ).
Transpose of matrix A, represented as ( A^T ), switches the arrangement of rows and columns.
Matrix operations observe associative and distributive laws:
- Associative: ( A(BC) = (AB)C ).
- Distributive: ( A(B + C) = AB + AC ).