#L05 · Linear Algebra & Manifolds · Lv.3View in 3D Mind-Map

Determinant & Matrix Inverse

Seki Kowa, Leibniz & Cauchy · 1683 CE

\det(A) = ad - bc,\quad A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

AA^{-1} = I,\quad \det(AB) = \det(A)\det(B)

Measures the signed area/volume scaling factor of a linear transformation via det(A), and reverses the transformation via A⁻¹ whenever det(A) ≠ 0.

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Entry a

Entry b

Entry c

Entry d

\det(A) = (3)(4) - (1)(2) = 10.00

행렬식 det(A) = ad-bc

10.0000

역행렬 존재 여부

가역 (Invertible)

A⁻¹의 (1,1) 성분 d/det

0.400

Prerequisite Path (13 STEP)

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Click steps in order to follow the 3D neon path from basics to target.

STEP 1#A02Polynomial Expansion & Factoring

(a+b)^2 = a^2 + 2ab + b^2

STEP 2#A03Laws of Exponents & Radicals

a^m \cdot a^n = a^{m+n}

STEP 3#G01Pythagorean Theorem

a^2 + b^2 = c^2 \implies c = \sqrt{a^2 + b^2}

STEP 4#G03Pi, Radian Measure & Circular Sector

l = r\theta

STEP 5#L01Vector Operations, L1/L2 Norms & Normalization

\|\mathbf{v}\|_2 = \sqrt{\sum_{i=1}^{n} v_i^2}

STEP 6#A01Linear Equation & Balance Axiom

ax + b = 0 \implies x = -\frac{b}{a}

STEP 7#G022D/3D Euclidean Distance Formula

d(P

STEP 8#G05Law of Sines & Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos C

STEP 9#G04Unit Circle Trigonometry & Identities

\cos^2\theta + \sin^2\theta = 1

STEP 10#L02Dot Product & Cosine Similarity

\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta

STEP 11#L033D Cross Product & Surface Normal Vector

\mathbf{a} \times \mathbf{b} = (a_y b_z - a_z b_y

STEP 12#L04Matrix Multiplication & Linear Transformation

(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}

STEP 13#L05Determinant & Matrix Inverse

\det(A) = ad - bc

🚀 Next Unlocks (Derived Formulas) (3)

#L06Gaussian Elimination & Linear Systems (Ax = b)#L072D/3D Rotation Matrix & Orthogonal Group SO(3)#L08Eigenvalues & Eigenvectors (Av = λv)

Geometric & Intuitive Insight

가로·세로가 1인 단위 정사각형에 행렬 A를 곱하면 두 열벡터 (a, c)와 (b, d)가 만드는 평행사변형으로 변형되는데, 그 넓이가 정확히 ad - bc 입니다! 넓이가 0이 되어 선분으로 찌그러지면 다시 정사각형으로 복구(역행렬)할 수 없습니다.

Step-by-Step Derivation

Step 1.1단계: 2×2 행렬 A의 두 열벡터 (a, c, 0)과 (b, d, 0)의 외적 z성분(평행사변형 부호 넓이)은 ad - bc 입니다.
Step 2.2단계: A · [[d, -b], [-c, a]] 를 행렬 곱셈하면 대각선은 ad - bc, 비대각선은 0이 됩니다.
Step 3.3단계: 따라서 ad - bc ≠ 0 일 때 양변을 (ad - bc)로 나누면 AA⁻¹ = I 를 만족하는 역행렬이 유도됩니다.

AI & Coding Real-World Use

3D mouse raycasting unprojection (MVP⁻¹) in games and Jacobian determinants in Normalizing Flow AI

#3D Mouse Raycasting#Jacobian Determinant#Normalizing Flows

const det2 = (a:number, b:number, c:number, d:number) => a * d - b * c;

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Matrix Multiplication & Linear Transformation

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