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Computational Linear Algebra: Matrix Theory, Transformations & Determinants

In modern computational mathematics, computer graphics, quantum mechanics, and machine learning, matrices are rectangular arrays of scalar real numbers $\mathbb{R}^{m \times n}$ representing linear maps between finite-dimensional vector spaces. Matrix algebra systematizes multivariate systems of simultaneous equations into concise operator equations $A\mathbf{x} = \mathbf{b}$.

Fundamental Matrix Operations

Let $A, B \in \mathbb{R}^{m \times n}$ share identical dimensional topology. Element-wise addition and scalar multiplication by $\lambda \in \mathbb{R}$ evaluate to:

(A \pm B)_{ij} = A_{ij} \pm B_{ij}, \qquad (\lambda A)_{ij} = \lambda \cdot A_{ij}

For matrix multiplication, the column cardinality of operator $A \in \mathbb{R}^{m \times k}$ must strictly equate to the row cardinality of operand $B \in \mathbb{R}^{k \times n}$, generating product $C \in \mathbb{R}^{m \times n}$ via inner Euclidean dot products:

C_{ij} = (AB)_{ij} = \sum_{p=1}^k A_{ip} B_{pj} = A_{i1}B_{1j} + A_{i2}B_{2j} + \dots + A_{ik}B_{kj}

The Determinant & Invertibility Criterion

The determinant $\det(A)$ is an intrinsic multilinear alternating scalar characterizing the hyper-volume scaling factor of the linear transformation. For a $2 \times 2$ matrix, the determinant evaluates to:

\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc

For a $3 \times 3$ matrix, expansion along the first row via Laplace cofactors yields:

\det \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)

A square matrix $A$ is invertible (non-singular) if and only if $\det(A) \neq 0$. The unique multiplicative inverse $A^{-1}$ satisfies $A A^{-1} = A^{-1} A = I_n$ and computes via the adjugate matrix $\text{adj}(A)$ (the transpose of the cofactor matrix $C^T$):

A^{-1} = \frac{1}{\det(A)} \text{adj}(A) = \frac{1}{\det(A)} C^T

Step-by-Step Matrix Inversion Example

Inverting the non-singular $2 \times 2$ matrix $A = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$:

Determinant: $\det(A) = (4 \times 6) - (7 \times 2) = 24 - 14 = \mathbf{10} \neq 0$ (Invertible).
Adjugate Construction: Swap diagonal elements ($4 \leftrightarrow 6$), negate off-diagonals ($-7, -2$): $$\text{adj}(A) = \begin{pmatrix} 6 & -7 \\ -2 & 4 \end{pmatrix}$$
Matrix Inverse: $$A^{-1} = \frac{1}{10} \begin{pmatrix} 6 & -7 \\ -2 & 4 \end{pmatrix} = \mathbf{\begin{pmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{pmatrix}}$$
Verification Proof: $$A A^{-1} = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix} \begin{pmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{pmatrix} = \begin{pmatrix} 2.4 - 1.4 & -2.8 + 2.8 \\ 1.2 - 1.2 & -1.4 + 2.4 \end{pmatrix} = \mathbf{\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}} = I_2$$

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📝 Step-by-Step Computational Derivation

Computational Linear Algebra: Matrix Theory, Transformations & Determinants

Fundamental Matrix Operations

The Determinant & Invertibility Criterion

Step-by-Step Matrix Inversion Example