Geometric Bites

Finding the Inverse of a 2x2 Matrix from Scratch This post shows a complete, step-by-step derivation of the inverse of a 2x2 matrix. Everything is expressed using stable, browser-safe ASCII formatting so the layout displays correctly on all devices and all templates. FIRST PART. Start with the matrix equation: A = [[a, b], [c, d]] A^(-1) = [[w, x], [y, z]] Goal: A * A^(-1) = I This produces the column equations: [aw + by, cw + dy]^T = [1, 0]^T [ax + bz, cx + dz]^T = [0, 1]^T Which gives the four equations: aw + by = 1 cw + dy = 0 ax + bz = 0 cx + dz = 1 SECOND PART. Use the first two equations to find w. aw + by = 1 cw + dy = 0 Multiply: (ad)w + (bd)y = d (first eq multiplied by d) (bc)w + (bd)y = 0 (second eq multiplied by b) Subtract: (ad - bc)w = d w = d / (ad - bc) (ad - bc != 0) THIRD PART. Use the next pair to find x. ax + bz = 0 cx + dz = 1 Multiply: (ad)x + (bd)z = 0 (bc)x + (bd)z = b Subtract: (ad - bc)...

Full Coordinate Derivation of (B - A) x (C - A) in R3 This derivation shows every algebraic step involved in expanding the cross product (B - A) x (C - A) using only coordinates. No vector identities are assumed in advance. All identities that appear at the end arise directly from the coordinate formula and elementary algebra. This method provides complete transparency and is the foundation for many geometric and analytic results involving the cross product. 1. Vectors and Cross Product Formula A = (a1, a2, a3) B = (b1, b2, b3) C = (c1, c2, c3) For vectors U = (u1, u2, u3) and V = (v1, v2, v3), the cross product is defined in coordinates by: U x V = ( u2*v3 - u3*v2, u3*v1 - u1*v3, u1*v2 - u2*v1 ) This is the only formula used. Every identity later in the derivation follows from substituting coordinates into this definition. 2. Basic Cross Products Needed for the Expansion A x A A x A = ( a2*a3 - a3*a2, a3*a1 - a1*a3, a1*a2 - a2*a1 ) = (0, 0, 0) A vector...