Chapter 2 · Lesson 3 · Strang §2.5, §2.7

Inverses, transposes and permutations

By the end of this lesson you can

Undo a matrix, and know when it can't be undone
Invert any 2 × 2 in your head, and bigger ones with Gauss-Jordan
Use transposes and permutations without second-guessing

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.

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Slides

Lesson 3The inverseThe inverse matrix: undo what A doesOpen slide →01 · Strang §2.5 · Inverse matrices

Inverse Matrix Explained Visually

The inverse matrix undoes A: apply A, then its inverse, and every vector is back where it started. Worked examples, moving pictures and Python. Strang §2.5.

Lesson 32 × 2 inverseThe 2 by 2 inverse: swap, flip signs, divide by ad − bcOpen slide →02 · Strang §2.5 · The 2 by 2 inverse

2×2 Inverse Matrix Formula and When It Fails

Swap the diagonal, flip the other signs, divide by ad − bc. What the determinant ad − bc tells you, and why 0 means no inverse. Worked examples and Python. Strang §2.5.

Lesson 3Inverse of ABInverse of a product: undo the steps in reverse orderOpen slide →03 · Strang §2.5 · Inverse of a product

Inverse of a Product: Why (AB)⁻¹ = B⁻¹A⁻¹

To undo AB you undo A first, then B, so the order flips: (AB)⁻¹ = B⁻¹A⁻¹. Socks and shoes, worked 2×2 and 3×3 examples and live Python. Strang §2.5.

Lesson 3Gauss-JordanGauss-Jordan: find A⁻¹ by elimination on [A | I]Open slide →04 · Strang §2.5 · Gauss-Jordan elimination

Gauss-Jordan Method: Find the Inverse of a Matrix

Put A next to the identity and eliminate until A becomes I. The identity turns into the inverse. Worked 2×2 and 3×3 examples, with rref in Python. Strang §2.5.

Lesson 3TransposeThe transpose: rows become columns, and symmetric matricesOpen slide →05 · Strang §2.7 · Transposes

Transpose and Symmetric Matrices Explained

The transpose swaps rows and columns. Why (AB)ᵀ = BᵀAᵀ, what a symmetric matrix is, and why AᵀA is always symmetric. Worked examples and Python. Strang §2.7.

Lesson 3PermutationsPermutation matrices: reorder the rowsOpen slide →06 · Strang §2.7 · Permutations

Permutation Matrices: Reordering Rows

Permutation matrices reorder the rows of a matrix. Why P⁻¹ = Pᵀ, and how PA = LU handles zero pivots, with worked 3×3 examples and Python. Strang §2.7.

Lesson 3RecapWhat you can now doOpen slide →07 · Recap · Reading

Lesson 3 Recap: Inverses, Transposes and Permutations

Lesson 3 in one page: inverses, the 2×2 formula and the determinant ad − bc, the inverse of a product, Gauss-Jordan, transposes, symmetric matrices and permutation matrices.