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)...

Reversing a Linear Transformation Using an Inverse Matrix In linear algebra, any invertible linear transformation can be reversed. The key tool that makes this possible is the inverse matrix . If a matrix transforms a vector into another, the inverse matrix recovers the original. 1. The Transformation Equation Suppose a vector x₁ is transformed into a vector x₂ using a matrix T : T x₁ = x₂ This equation describes how x₁ is mapped to x₂ . To reverse the transformation, we must apply the inverse matrix. 2. Applying the Inverse Matrix Multiply both sides of the equation by T⁻¹ : T⁻¹ (T x₁) = T⁻¹ x₂ Using the fundamental identity: T⁻¹ T = I the expression simplifies directly to: x₁ = T⁻¹ x₂ 3. Interpretation This tells us that the original vector is obtained by applying the inverse matrix to the transformed vector: Original vector = Inverse matrix × Image vector As long as the matrix is invertible, the reverse transformation always exist...