Final Review Sheet for Math 207, Spring 2007

Concepts to know from Chapter 2

  1. What is the determinant det(A) of an n×n matrix A?
    1. the recursive definition
      1. what is a minor
      2. what is a cofactor
      3. what is the cofactor expansion along a row or column
    2. special cases
      1. 1×1 matrices
      2. 2×2 matrices
      3. upper-triangular matrices
      4. lower-triangular matrices
      5. diagonal matrices
      6. (other) matrices with very few non-zero entries
      7. matrices with a row or column of zeros
      8. matrices with a row (or column) that is a multiple of another row (or column)
    3. the combinatorial definition
      1. what are permutations of the numbers 1,2,3,...,n
      2. the number of inversions in a permutation
      3. the sign of a permuation
      4. what are elementary products from an n×n matrix A
      5. det(A) as the sum of all elementary products from A, each with the corresponding sign of its permutation
  2. what determinants are for
    1. MOST IMPORTANT FACT: det(A) = 0 if and only if A is singular
    2. Cramer's Rule
    3. the adjoint formula for the inverse of a matrix
      1. what is the adjoint adj(A) of a matrix A
      2. a formula for A-1 in terms of det(A) and adj(A)
  3. changing a matrix, seeing the corresponding change of its determinant
    1. det(AT)=det(A)
    2. the effect of each type of elementary row operation on the determinant
    3. det(k A), where k is a scalar and A a square matrix
    4. det(A-1)
  4. an incredibly useful relationship: det(AB)=det(A)det(B)
  5. eigenvalues and eigenvectors
    1. what they are
    2. the equation to solve for eigenvalues: det(A-λI)=0
    3. the equation to solve for eigenvectors: (A-λI)X=0, for X a non-trivial vector
  6. VERY IMPORTANT: add the condition det(A)≠0 as another equivalent version of the listed equivalent statements in the big theorem on properties of invertible matrices (for which, see the Midterm II Review Sheet, "Concept Q" or the book's Theorem 2.3.6 on page 109.

Things to know how to do

  1. compute the determinant
    1. with the recursive definition
    2. with the combinatorial definition
    3. by row-reducing the matrix, tracking the corresponding changes to the determinant, and finally computing the determinant of a triangular matrix
    4. in special cases (2×2, 3×3, UT, LT, diagonal, sparse, with zero rows/columns or rows/columns that are related to other rows/columns)
  2. use the value of the determinant as a hypothesis or conclusion in applications of the "big theorem on invertible matrices" mentioned above
  3. find eigenvalues and eigenvectors
  4. find solutions of linear systems with Cramer's rule
  5. use the adjoint formula for the inverse of a matrix
  6. track the changes in the determinant corresponding to changes in the matrix (transpose, row or column operations, inversion, etc.)


Jonathan Poritz (jonathan.poritz@gmail.com)