Multivariable Calculus

Vectors and Dot Products

01Vectors, Components, and Length in Space 02Scaling and Adding Vectors: The Parallelogram Rule 03The Dot Product and the Law of Cosines 04Finding Angles in Space With the Dot Product 05The Perpendicularity Test and the Normal Vector to a Plane Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

Determinants and the Cross Product

01The Component of a Vector Along Any Unit Direction 02The 2x2 Determinant and the Area of a Parallelogram 03The 3x3 Determinant and the Volume of a Box 04The Cross Product and the Right-Hand Rule 05The Triple Product and the Volume of a Parallelepiped Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

Matrices and Inverse Matrices

01Why Swapping a Cross Product Flips Its Sign 02The Equation of a Plane Through Three Points 03Matrix Multiplication: Rows Dotted With Columns 04Matrices as Transformations: Identity and Rotation 05The Inverse Matrix and Solving AX = B 06Computing a 3x3 Inverse: Minors, Cofactors, Adjoint Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

Planes and Square Systems

01Planes From a Point and a Normal Vector 02Reading the Normal Vector Off a Plane Equation 03Three Planes, One Point: Solving a 3x3 System 04Invertibility, det A, and the Homogeneous System 05When det A = 0: Coplanar Normals and a Line of Solutions Problem set0/10 Problem set 20/10 Practice∞

The Inverse Matrix and Solving AX = B

Why must A inverse multiply AX = B from the left, and what goes wrong the moment you put it on the right?

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Your summary note

1. The defining property of the inverse $A^{-1}$
  Write the two defining equations $AM = I$ and $MA = I$ , record that verifying either one suffices, and state that $M$ undoes $A$ and reverses the change of variables.
2. Square matrices and the notation $A^{-1}$
  State that only a square $n \times n$ matrix can have an inverse, and record $A^{-1}$ as the notation for that matrix.
3. The formula $X = A^{-1}B$
  Reproduce the derivation: multiply both sides of $AX = B$ on the left by $A^{-1}$ , simplify $(A^{-1}A)X$ to $IX$ , and write the resulting formula.
4. The left-multiplication rule and the mismatch of $XA$
  Record that $A^{-1}$ goes on the left of both sides, and that with $X$ a column vector $AX$ is defined while $XA$ is not.
Attempt 1 of 2