# 统计代写|主成分分析代写Principal Component Analysis代考|”Understanding the Determinant of a Square Matrix: Calculation and Concepts”

The determinant of a square matrix is a unique scalar value associated with that particular matrix. It is mathematically represented as det[A] or |A| for an n×nn \times nn×n matrix:

det[A]=∣A∣=∣a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮an1an2⋯ann∣\text{det}[A] = |A| = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} a_{21} & a_{22} & \cdots & a_{2n} \vdots & \vdots & \ddots & \vdots a_{n1} & a_{n2} & \cdots & a_{nn} \end{vmatrix} det[A]=∣A∣= ​

a 11 ​

a 21 ​

⋮ a n1 ​

a 12 ​

a 22 ​

⋮ a n2 ​

⋯ ⋯ ⋱ ⋯ ​

a 1n ​

a 2n ​

⋮ a nn ​

To calculate the determinant, there is a systematic process depending on the size of the matrix. For a 2 × 2 matrix,

[A]=[a11a12a21a22],[A] = \begin{bmatrix} a_{11} & a_{12} a_{21} & a_{22} \end{bmatrix}, [A]=[ a 11 ​

a 21 ​

a 12 ​

a 22 ​

​ ], its determinant is given by:

∣A∣=a11a22−a12a21.|A| = a_{11}a_{22} – a_{12}a_{21}.∣A∣=a 11 ​ a 22 ​ −a 12 ​ a 21 ​ .

Expanding this concept to a 3 × 3 matrix,

[A]=∣a11a12a13a21a22a23a31a32a33∣,[A] = \begin{vmatrix} a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33} \end{vmatrix}, [A]= ​

a 11 ​

a 21 ​

a 31 ​

a 12 ​

a 22 ​

a 32 ​

a 13 ​

a 23 ​

a 33 ​

​ , its determinant is computed using the formula:

∣A∣=a11(a22a33−a23a32)−a12(a21a33−a23a31)+a13(a21a32−a22a31).|A| = a_{11}(a_{22}a_{33} – a_{23}a_{32}) – a_{12}(a_{21}a_{33} – a_{23}a_{31}) + a_{13}(a_{21}a_{32} – a_{22}a_{31}).∣A∣=a 11 ​ (a 22 ​ a 33 ​ −a 23 ​ a 32 ​ )−a 12 ​ (a 21 ​ a 33 ​ −a 23 ​ a 31 ​ )+a 13 ​ (a 21 ​ a 32 ​ −a 22 ​ a 31 ​ ).

Each term inside the parentheses represents the determinant of a smaller, 2 × 2 matrix obtained by deleting the first row and one of the columns. These are called minors, denoted by ∣Mij∣|M_{ij}|∣M ij ​ ∣. The cofactor CijC_{ij}C ij ​ of an element aija_{ij}a ij ​ is the minor ∣Mij∣|M_{ij}|∣M ij ​ ∣ multiplied by (−1)i+j(-1)^{i+j}(−1) i+j , where the sign alternates based on the sum of the row and column indices:

Cij=(−1)i+jMij.C_{ij} = (-1)^{i+j} M_{ij}.C ij ​ =(−1) i+j M ij ​ .

The determinant can then be expanded in terms of these cofactors, for instance, expanding along the first row:

∣A∣=a11C11+a12C12+a13C13.|A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}.∣A∣=a 11 ​ C 11 ​ +a 12 ​ C 12 ​ +a 13 ​ C 13 ​ .

This expansion principle applies to any row or column of the matrix, leading to the more general formulas:

∣A∣=∑j=1naijCij,|A| = \sum_{j=1}^{n} a_{ij}C_{ij},∣A∣=∑ j=1 n ​ a ij ​ C ij ​ , ∣A∣=∑i=1naijCij.|A| = \sum_{i=1}^{n} a_{ij}C_{ij}.∣A∣=∑ i=1 n ​ a ij ​ C ij ​ .

For a matrix of any order, the process continues until all remaining minors are of order 2, which can then be calculated using the 2 × 2 determinant rule.

The inverse of a square matrix [A][A][A], denoted by [A]−1[A]^{-1}[A] −1 , is another square matrix such that:

[A]−1[A]=[A][A]−1=[I],[A]^{-1}[A] = [A][A]^{-1} = [I],[A] −1 [A]=[A][A] −1 =[I],

where [I][I][I] is the identity matrix of order nnn. The inverse is pivotal for solving systems of simultaneous linear equations using matrix methods. Given the system:

a11x1+a12x2+⋯+a1nxn=y1a21x1+a22x2+⋯+a2nxn=y2⋮an1x1+an2x2+⋯+annxn=yn,\begin{align*} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n &= y_1 a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n &= y_2 &\vdots a_{n1}x_1 + a_{n2}x_2 + \cdots + a_{nn}x_n &= y_n, \end{align*} a 11 ​ x 1 ​ +a 12 ​ x 2 ​ +⋯+a 1n ​ x n ​

a 21 ​ x 1 ​ +a 22 ​ x 2 ​ +⋯+a 2n ​ x n ​

a n1 ​ x 1 ​ +a n2 ​ x 2 ​ +⋯+a nn ​ x n ​

=y 1 ​

=y 2 ​

⋮ =y n ​ , ​

it can be rewritten compactly as:

[A]x=y.[A]\mathbf{x} = \mathbf{y}.[A]x=y.

Finding [A]−1[A]^{-1}[A] −1 allows us to express the solution x\mathbf{x}x directly as x=[A]−1y\mathbf{x} = [A]^{-1}\mathbf{y}x=[A] −1 y.

### MATLAB代写

MATLAB 是一款高性能的技术计算语言，集成了计算、可视化和编程环境于一体，以熟悉的数学符号表达问题和解决方案。MATLAB 的基本数据元素是一个不需要维度的数组，使得能够快速解决带有矩阵和向量公式的多种技术计算问题，相比使用 C 或 Fortran 等标量非交互式语言编写的程序，效率大大提高。MATLAB 名称源自“矩阵实验室”（Matrix Laboratory）。最初开发 MATLAB 的目标是为了提供对 LINPACK 和 EISPACK 项目的矩阵软件的便捷访问，这两个项目代表了当时矩阵计算软件的先进技术。经过长期发展和众多用户的贡献，MATLAB 已成为数学、工程和科学入门及高级课程的标准教学工具，在工业界，MATLAB 是高效研究、开发和分析的理想选择。MATLAB 提供了一系列名为工具箱的特定应用解决方案集，这对广大 MATLAB 用户至关重要，因为它们极大地扩展了 MATLAB 环境，使其能够解决特定类别问题。工具箱包含了针对特定应用领域的 MATLAB 函数（M 文件），涵盖信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等诸多领域。