Matrices & Determinants

1. Introduction

A matrix is a rectangular array of numbers arranged in rows and columns, and it is arguably the single most useful object in applied mathematics. Matrices encode systems of linear equations, represent linear transformations of space (rotations, scalings, shears, projections), store and manipulate data, drive computer graphics and machine learning, and model networks, Markov chains, and dynamical systems. The determinant is a single number distilled from a square matrix that captures whether the matrix is invertible and how it scales area or volume.

The power of matrices comes from a small algebra: we can add them, scale them, multiply them, transpose them, and — when possible — invert them. Each operation has a precise definition and clean properties, but also a few famous traps (matrix multiplication is not commutative; the inverse exists only sometimes). The determinant ties everything together: a square matrix is invertible exactly when its determinant is nonzero, and the determinant of a product is the product of determinants.

This article develops the theory rigorously: matrix operations and their algebraic laws, the geometric meaning of matrices as linear maps, determinants via cofactor expansion and row operations, the inverse via the adjugate and via Gaussian elimination, elementary row operations, and applications to solving linear systems (including Cramer's rule). The goal is both computational fluency and conceptual understanding of what these objects mean.

Matrices are the computational backbone of linear algebra: every system of linear equations, every linear transformation, and every change of coordinates reduces to matrix operations. The determinant is the bridge between algebra (invertibility) and geometry (volume scaling), and understanding both viewpoints is essential for the deeper theory of eigenvalues, vector spaces, and the spectral theorem.

2. Core Concepts

2.1 Matrices and Basic Operations

An $m\times n$ matrix $A$ has $m$ rows and $n$ columns, with entry $a_{ij}$ in row $i$, column $j$. Addition and scalar multiplication act entrywise (requiring matching dimensions for addition): $(A+B)_{ij}=a_{ij}+b_{ij}$ and $(cA)_{ij}=c\,a_{ij}$. The transpose $A^{T}$ swaps rows and columns: $(A^{T})_{ij}=a_{ji}$.

2.2 Matrix Multiplication

The product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$: if $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is $m\times p$ with $(AB)_{ij} = \sum_{k=1}^{n} a_{ik}\,b_{kj}.$ Multiplication is associative $(AB)C=A(BC)$ and distributive $A(B+C)=AB+AC$, but not commutative: in general $AB\ne BA$. The identity matrix $I_n$ satisfies $AI=IA=A$.

Viewing $A$ as a linear map $\mathbf{x}\mapsto A\mathbf{x}$, the product $AB$ means "apply $B$, then $A$" — another reason order matters.

2.3 The Determinant: Definition

For a $2\times2$ matrix: $\det\begin{bmatrix} a & b \\ c & d\end{bmatrix} = ad - bc.$ For larger matrices, cofactor (Laplace) expansion along any row $i$: $\det(A) = \sum_{j=1}^{n} (-1)^{i+j}\,a_{ij}\,M_{ij},$ where $M_{ij}$ is the minor (determinant of the submatrix with row $i$, column $j$ deleted) and $C_{ij}=(-1)^{i+j}M_{ij}$ is the cofactor.

2.4 Geometric Meaning of the Determinant

The determinant measures how a linear transformation scales signed area (2D) or volume (3D). If $A$ maps the unit square to a parallelogram, that parallelogram's area is $|\det(A)|$; the sign records whether orientation is preserved ($+$) or flipped ($-$). A determinant of zero means the transformation collapses space to a lower dimension — the columns are linearly dependent — which is precisely why $\det(A)=0$ signals non-invertibility.

2.5 Key Determinant Properties

• $\det(I)=1$.
• $\det(AB)=\det(A)\det(B)$ (multiplicativity).
• $\det(A^{T})=\det(A)$.
• $\det(A^{-1})=1/\det(A)$ when $A$ is invertible.
• $\det(cA)=c^{n}\det(A)$ for an $n\times n$ matrix.
• Swapping two rows multiplies the determinant by $-1$; adding a multiple of one row to another leaves it unchanged; scaling a row by $c$ multiplies the determinant by $c$.
• A triangular matrix's determinant is the product of its diagonal entries.

Proof sketch of $\det(AB)=\det(A)\det(B)$. The determinant is the unique function on $n\times n$ matrices that is multilinear in columns, alternating (swapping columns negates it), and normalized to $\det(I)=1$. Multilinearity and alternation force $\det$ to be the signed volume scaling factor. Since $AB$ as a column map is composition, the volume scales by $\det(A)\det(B)$.

2.6 Elementary Row Operations and Row Echelon Form

The three elementary row operations are: swap two rows, multiply a row by a nonzero scalar, add a multiple of one row to another. Each corresponds to left-multiplication by an elementary matrix. Row reduction to echelon form (and reduced row echelon form, RREF) is the backbone of solving systems and computing rank. The determinant tracks how each operation affects its value.

2.7 The Inverse Matrix

The inverse $A^{-1}$ satisfies $AA^{-1}=A^{-1}A=I$. It exists iff $\det(A)\ne 0$. For $2\times2$: $A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a\end{bmatrix}.$ In general, $A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A)$, where $\operatorname{adj}(A)$ is the transpose of the cofactor matrix. Properties: $(A^{-1})^{-1}=A$, $(AB)^{-1}=B^{-1}A^{-1}$, $(A^{T})^{-1}=(A^{-1})^{T}$.

2.8 Solving Linear Systems

$A\mathbf{x}=\mathbf{b}$ with nonsingular $A$ has unique solution $\mathbf{x}=A^{-1}\mathbf{b}$. Cramer's rule: $x_i = \frac{\det(A_i)}{\det(A)},$ where $A_i$ is $A$ with column $i$ replaced by $\mathbf{b}$. For larger systems, Gaussian elimination is far more efficient.