Matrix Algebra & Determinants

1. The Matrix Ring

The collection of $m \times n$ matrices with entries in a field $\mathbb{F}$ ($\mathbb{R}$ or $\mathbb{C}$), denoted $M_{m \times n}(\mathbb{F})$, forms a vector space under entrywise addition and scalar multiplication.
For $A \in M_{m \times p}(\mathbb{F})$ and $B \in M_{p \times n}(\mathbb{F})$, their matrix product $C = AB \in M_{m \times n}(\mathbb{F})$ has entries: $$\mathbf{c_{ij} = \sum_{k=1}^p a_{ik} b_{kj}}$$ The product is associative ($A(BC) = (AB)C$) and distributes over addition, but is strictly non-commutative ($AB \ne BA$ in general). The square matrices $M_n(\mathbb{F})$ form a non-commutative ring with identity $I_n$.

2. The Transpose & Conjugate Transpose

The transpose $A^T$ of $A = [a_{ij}]$ is defined by $(A^T)_{ij} = a_{ji}$. Key properties: $$(A + B)^T = A^T + B^T, \quad (cA)^T = c A^T, \quad (A^T)^T = A, \quad \mathbf{(AB)^T = B^T A^T}$$ For complex matrices, the Hermitian adjoint (conjugate transpose) is $A^\dagger \equiv (\bar{A})^T$, satisfying $(AB)^\dagger = B^\dagger A^\dagger$.

3. The Trace Function

For a square matrix $A \in M_n(\mathbb{F})$, the trace is the sum of its diagonal elements: $$\mathbf{\text{tr}(A) \equiv \sum_{i=1}^n a_{ii}}$$ Cyclic Invariance Theorem: For any $A \in M_{m \times n}$ and $B \in M_{n \times m}$: $$\mathbf{\text{tr}(AB) = \text{tr}(BA)}$$ Proof: $\text{tr}(AB) = \sum_{i=1}^m (AB)_{ii} = \sum_{i=1}^m \sum_{j=1}^n a_{ij} b_{ji} = \sum_{j=1}^n \sum_{i=1}^m b_{ji} a_{ij} = \sum_{j=1}^n (BA)_{jj} = \text{tr}(BA)$. $\blacksquare$

1. Real Special Matrices

Let $A \in M_n(\mathbb{R})$:

• Symmetric: $A^T = A \iff a_{ij} = a_{ji}$.
• Skew-Symmetric: $A^T = -A \iff a_{ij} = -a_{ji}$ (diagonal entries must be zero: $a_{ii} = 0$).
• Orthogonal: $A^T A = A A^T = I \iff A^{-1} = A^T$. Rows (and columns) form an orthonormal basis of $\mathbb{R}^n$. Orthogonal transformations preserve Euclidean lengths and angles: $\|Ax\| = \|x\|$.

2. Complex Special Matrices

Let $A \in M_n(\mathbb{C})$:

• Hermitian: $A^\dagger = A \iff a_{ij} = \bar{a}_{ji}$ (diagonal entries must be purely real: $a_{ii} \in \mathbb{R}$).
• Skew-Hermitian: $A^\dagger = -A \iff a_{ij} = -\bar{a}_{ji}$ (diagonal entries must be purely imaginary).
• Unitary: $A^\dagger A = I \iff A^{-1} = A^\dagger$. Unitary matrices preserve the complex inner product: $\langle Ux, Uy \rangle = \langle x, y \rangle$.

3. Algebraic Operational Classes

• Idempotent: $A^2 = A$ (represents projection operators).
• Nilpotent: $A^k = 0$ for some integer $k \ge 1$ (the smallest such $k$ is the index of nilpotency).
• Involutory: $A^2 = I \iff A^{-1} = A$ (represents reflection operators).

1. Axiomatic Definition of the Determinant

The determinant is the unique function $\det: M_n(\mathbb{F}) \to \mathbb{F}$ satisfying three fundamental axioms:

1. Multilinearity: $\det$ is a linear function of each row when all other rows are held fixed.
2. Alternating Property: Swapping any two rows negates the determinant: $\det(R_1, \dots, R_i, \dots, R_j, \dots, R_n) = -\det(R_1, \dots, R_j, \dots, R_i, \dots, R_n)$. Consequently, if two rows are identical, $\det A = 0$.
3. Normalization: $\det(I_n) = 1$.

2. The Leibniz Permutation Formula

From the axioms, the explicit determinant formula is: $$\mathbf{\det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^n a_{i, \sigma(i)}}$$ where the sum ranges over all $n!$ permutations $\sigma$ of the symmetric group $S_n$, and $\text{sgn}(\sigma) \in \{+1, -1\}$ is the permutation parity.

3. Laplace's Cofactor Expansion Theorem

The $(i, j)$-th minor $M_{ij}$ is the determinant of the $(n-1) \times (n-1)$ submatrix formed by deleting row $i$ and column $j$. The $(i, j)$-th cofactor is: $$C_{ij} \equiv (-1)^{i+j} M_{ij}$$ Theorem (Laplace): The determinant can be evaluated by expanding along any arbitrary row $i$ or column $j$: $$\mathbf{\det(A) = \sum_{j=1}^n a_{ij} C_{ij} \quad (\text{Expansion along row } i)}$$ $$\mathbf{\det(A) = \sum_{i=1}^n a_{ij} C_{ij} \quad (\text{Expansion along column } j)}$$

1. Invariance and Operational Properties

• Transpose Invariance: $\det(A^T) = \det(A)$. Row operations and column operations have identical effects on determinants!
• Triangular Matrices: If $A$ is upper-triangular, lower-triangular, or diagonal, its determinant is simply the product of its diagonal entries: $$\det(A) = a_{11} a_{22} \cdots a_{nn}$$
• Type III Row Operations: Adding a scalar multiple of one row to another preserves the determinant strictly unchanged: $\det(A) = \det(E A)$ where $\det(E) = 1$.
• Scalar Multiplication: For an $n \times n$ matrix, $\det(c A) = c^n \det(A)$.

2. The Multiplicative Theorem

Theorem (Cauchy–Binet): For any two $n \times n$ matrices $A$ and $B$: $$\mathbf{\det(AB) = \det(A) \det(B)}$$ Immediate Corollaries:

• A matrix $A$ is invertible if and only if $\det(A) \ne 0$.
• If $A$ is invertible: $\det(A^{-1}) = \frac{1}{\det(A)}$.
• If $A$ and $B$ are similar ($B = P^{-1} A P$): $\det(B) = \det(P^{-1})\det(A)\det(P) = \det(A)$.
• If $Q$ is orthogonal: $Q^T Q = I \implies [\det(Q)]^2 = 1 \implies \det(Q) = \pm 1$.

1. The Classical Adjugate (Adjoint)

The adjugate $\text{adj}(A)$ of an $n \times n$ matrix $A$ is the transpose of its cofactor matrix $C = [C_{ij}]$: $$\mathbf{\text{adj}(A) \equiv C^T \implies [\text{adj}(A)]_{ij} = C_{ji} = (-1)^{i+j} M_{ji}}$$

2. The Fundamental Inversion Theorem

Theorem: For any $n \times n$ matrix $A$: $$\mathbf{A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = (\det A) I_n}$$ Proof: The $(i, k)$-th entry of $A \cdot \text{adj}(A)$ is $\sum_{j=1}^n a_{ij} [\text{adj}(A)]_{jk} = \sum_{j=1}^n a_{ij} C_{kj}$. If $i = k$, this is the Laplace expansion of $\det(A)$ along row $i$. If $i \ne k$, this represents the expansion of a matrix with two identical rows ($i$ and $k$), which vanishes identically (alien cofactors). Thus $[A \cdot \text{adj}(A)]_{ik} = \delta_{ik} \det(A)$. $\blacksquare$

Consequently, whenever $\det(A) \ne 0$, the unique matrix inverse is: $$\mathbf{A^{-1} = \frac{1}{\det(A)} \text{adj}(A)}$$

3. Cramer’s Rule for Linear Systems

For an $n \times n$ system $AX = B$ with $\det(A) \ne 0$, $X = A^{-1} B = \frac{1}{\det A}\text{adj}(A) B$. Entrywise: $$\mathbf{x_i = \frac{\det(A_i)}{\det(A)}}$$ where $A_i$ is the matrix obtained by replacing the $i$-th column of $A$ with the constant vector $B$.