Recap · Reading
What you can now do.
Undo a matrix, find its inverse two ways, and use the transpose and permutations with confidence.
A⁻¹A = I (AB)⁻¹ = B⁻¹A⁻¹ (AB)ᵀ = BᵀAᵀ P⁻¹ = Pᵀ
- §2.5 The inverse. Rewind: A⁻¹ undoes A. It exists exactly when A has n pivots and det A ≠ 0.
- §2.5 The 2 by 2 formula. Swap the diagonal, flip the other signs, divide by ad − bc.
- §2.5 Products. Shoes off before socks: (AB)⁻¹ = B⁻¹A⁻¹.
- §2.5 Gauss-Jordan. Carbon paper: eliminate on [A | I] until it reads [I | A⁻¹].
- §2.7 Transposes. Turn the spreadsheet. Symmetric matrices are distance tables, and AᵀA is always one.
- §2.7 Permutations. A new playlist order. P⁻¹ = Pᵀ, and row exchanges give PA = LU.
Next lesson: the four subspaces (chapter 3)
What happens when A is not invertible, or not even square? The nullspace, pivot and free columns, the complete solution to Ax = b, and the big picture that ties it all together.
Read along
Gilbert Strang, Introduction to Linear Algebra, 5th edition (2016), sections 2.5 and 2.7. Lectures 3 and 5 of MIT 18.06 on MIT OpenCourseWare.