Summary linear algebra (notes, exercises, correction)

CHAPTER 3 DETERMINANTS

3.1 Determinants


• The determinant of a 2 × 2 matrix is
 
a b
det   = ad − bc
c d


 
2 3
• Example Find the determinant of  .
4 6




• Given an n × n matrix A, the matrix Aij is an (n − 1) × (n − 1) matrix obtained
by deleting the i row and j column of A.




1

,2

• Determinants


For n ≥ 2, the determinant of the n × n matrix A = [aij ] is given by

det A =a11 det A11 − a12 det A12 + . . . + (−1)n+1 a1n det A1n
n
X
= (−1)1+j a1j det A1j
j=1




 
1 5 0
 
• Example Compute the determinant of  3 −2 −1 .
 
 
4 7 −2

, 3

• Cofactor

Let A be an n × n matrix. The cofactor Cij is defined as

Cij = (−1)i+j det Aij




• Theorem (Laplace Expansion for the determinant) (Proof is omitted.)

The determinant of an n × n matrix A can be computed by a cofactor ex-
pansion across any row or any column. Namely
n
X
det A = aij Cij
j=1
n
X
= aij Cij
i=1

, 4