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∞

Computing a 3x3 Inverse: Minors, Cofactors, Adjoint

What do minors, a checkerboard of signs, a transpose, and one division by the determinant do to hand you A inverse?

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Your summary note

1. The four steps behind $A^{-1} = \frac{1}{\det(A)}\operatorname{adj}(A)$
  Write the formula down, name the four steps in order, and build the matrix of minors of the worked $3 \times 3$ example by deleting each entry's row and column.
2. The checkerboard sign rule for cofactors
  Draw the $+\,-\,+$ checkerboard, record that a plus leaves an entry alone and a minus flips its sign rather than forcing a positive or negative, and write out the cofactor matrix.
3. Transposing to $\operatorname{adj}(A)$ , then dividing by $\det(A)$
  Define the adjoint as the transpose of the cofactor matrix, note that the divisor is $\det(A)$ and never the determinant of the adjoint, and finish the worked example.
4. The two uses of $A^{-1}$
  Record that $A^{-1}$ reverses the change of variables as $X = A^{-1}U$ , and that it solves a linear system $AX = B$ as $X = A^{-1}B$ .
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