Determinants (10/17/05)

1
Determinants (10/17/05)

• We learned previously the determinant of the 2 by
  2 matrix A is the number a d b c .
• We need now to learn how to compute the
  determinant of larger square matrices.
• The definition is recursive, meaning we define
  the determinant of an n by n matrix in terms of
  the determinants of n different n 1 by n 1
  matrices.

2
Definition of Ai j and cofactors

• If A is an n by n matrix, then by Ai j we mean
  the n 1 by n 1 matrix obtained by removing
  the i th row and the j th column of A .
• By the (i, j )-cofactor Ci j of A , we mean the
  number (-1)i j det(Ai j )
• The (-1)i j causes the signs to alternate.

3
Definition of Det(A)

• If A is the n by n matrix ai j with cofactors
  Ci j , then for any fixed row i,det(A) ai 1
  Ci 1 ai 2 Ci 2 ai n Ci n
• One can also fix any column j .
• No matter which row or column you use, the same
  answer will emerge.
• Hence it makes sense to seek out a row or column
  with as many zeros as possible.

4
Some Special Cases

• If A has a row or column of all zeros, then its
  determinant must be zero
• A square matrix is called upper triangular if all
  the entries below the main diagonal are 0.
  Similarly lower triangular.
• The determinant of an upper or lower triangular
  matrix is just the product of the entries on the
  main diagonal.

5
Assignment for Wednesday

• Work on test corrections (due Friday).
• Read Section 3.1 (Note We shall return to some
  of the applications in Chapter 2, but no more of
  the regular material.)
• Do the Practice and Exercises 1 13 odd, 19, 21,
  and 23.
• In addition, explore the possible truth of the
  following two formulas (look at examples)det(A
  B) det(A) det(B) ?det(A B) det(A) det(B)
  ?