The matrix that undoes A, when it exists, how Gauss-Jordan finds it, what the transpose does, and the matrices that reorder rows. 7 short slides, then worked examples and problems. Each slide has an everyday picture, a worked example and Python you can run right on the page.
The inverse matrix undoes A: apply A, then its inverse, and every point is back where it started. See it on the plane and build one in Python. Strang §2.5.
02 · Strang §2.5 · The 2 by 2 inverseSwap the diagonal, flip the other signs, divide by ad − bc. Why the determinant appears, and why det = 0 means no inverse. Python example.
03 · Strang §2.5 · Inverse of a productTo undo AB you undo A first, then B, so the order flips: (AB)⁻¹ = B⁻¹A⁻¹. Socks and shoes, a worked 2×2 example and live Python. Strang §2.5.
04 · Strang §2.5 · Gauss-Jordan eliminationPut A next to the identity and eliminate until A becomes I. The identity turns into the inverse. Worked 3×3 example with rref in Python. Strang §2.5.
05 · Strang §2.7 · TransposesThe transpose swaps rows and columns. Why (AB)ᵀ = BᵀAᵀ, what a symmetric matrix is, and why AᵀA is always symmetric. Python example. Strang §2.7.
06 · Strang §2.7 · PermutationsA permutation matrix reorders the rows of A. Its inverse is its transpose, and it is how elimination handles row exchanges: PA = LU. Python example. Strang §2.7.
07 · Recap · ReadingInverses, the 2×2 formula, the inverse of a product, Gauss-Jordan, transposes, symmetric matrices and permutation matrices on one page.
✎ · PracticeWorked examples and practice problems with hidden solutions: 2×2 inverses, Gauss-Jordan, inverse of a product, transposes, symmetric and permutation matrices.