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∞

Invertibility, det A, and the Homogeneous System

Why does one number, det A, decide whether a matrix can be inverted at all — and why does the origin become the only solution?

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Your summary note

1. The one step of $A^{-1}$ that can fail
  Write $A^{-1} = \frac{1}{\det A}\operatorname{adj}(A)$ , record that the adjoint is always computable, and identify the final division as the step that breaks down.
2. The exact criterion for an inverse to exist
  State the theorem in both directions, record that a zero determinant leaves no inverse at all rather than one failed method, and name the matching picture of three planes.
3. The trivial solution of $AX = \mathbf{0}$
  Write the homogeneous version of the example system, record that its three planes all pass through the origin, and carry $X = A^{-1}\mathbf{0} = \mathbf{0}$ through to uniqueness.
Attempt 1 of 2