Final Review Sheet for Math 207, Spring 2009

Concepts we covered in Chapters 3 & 4:

  1. algebra of matrices
    1. addition
    2. scalar multiplication
    3. matrix multiplication
    4. interactions between operations
      1. associativity
      2. distributivity
      3. matrix multiplication is not commutative
    5. the transpose of a matrix
      1. from which a symmetric matrix
      2. and also an anti-symmetric matrix
    6. the inverse A-1 of an n×n matrix A
      1. definition, basic facts (mostly A A-1 = A-1 A = I)
      2. the words invertible [= "the inverse exists"] and singular [= "no inverse exists"]
      3. the formula for 2×2 matrices
      4. inverses of elementary matrices
      5. finding inverses in general: the Gauss-Jordan method
      6. [using a calculator to find an inverse of a numerical matrix]
      7. properties:
        1. (A B)-1 = B-1 A-1
        2. (c A)-1 = (1/c) A-1
      8. the "Fundamental Theorem on Invertible Matrices" — the following are equivalent:
        1. A is invertible
        2. the RREF form of A is the identity
        3. A X = B (for vectors X and B) has a unique solution for any B
        4. A X = 0 has only the trivial solution X=0
        5. A (and A-1) can be written as a product of elementary matrices
  2. eigenvectors and eigenvalues
    1. definitions: A v = λ v for v a non-zero vector
    2. equivalently: v is a nontrivial solution to the homogenous system with coefficient matrix A - λI
  3. determinants
    1. the definition by cofactor expansion along a row or column
    2. special cases
      1. 2×2 matrices
      2. upper-triangular matrices
      3. lower-triangular matrices
      4. diagonal matrices
      5. elementary matrices
      6. matrices with a row or column of zeros
    3. MOST IMPORTANT FACT: det(A) = 0 if and only if A is singular
    4. an incredibly useful relationship: det(AB)=det(A)det(B)
  4. VERY IMPORTANT: add the condition det(A)≠0 as another equivalent version of the listed equivalent statements in the fundamental theorem at A.6.h, above.

Things to know how to do

  1. manipulate products of matrices (and of their inverses and determinants) algebraically, watching out for non-commutativity
  2. invert a matrix
    1. with a calculator, for numerical matrices
    2. with the formula for 2×2 matrices
    3. using Gauss-Jordan
    4. for special kinds of matrices
  3. compute determinants in various ways
    1. with a calculator, for numerical matrices
    2. with the formula for 2×2 matrices
    3. for special kinds of matrices
  4. find eigenvalues and eigenvectors
  5. find solutions of linear systems by computing X = A-1 B
  6. write an invertible matrix (or its inverse) as a product of elementary matrices
  7. be able to explain the "how" and the "why" for any of the above