Matrices

1
Matrices

• We have studied several methods for solving a
  linear system of two equations in two unknowns,
  such as
• This coefficients of this system can be displayed
  as a matrix, which is a rectangular array of
  numbers arranged in rows and columns.
  Thus,
  
  is the coefficient matrix
  associated with the given system.

rows
columns
2
More about Matrices

• The general linear system of two equations in two
  unknowns can be written as
• Associated with this general system is the
  coefficient matrix
• We say that A is a 2?2 matrix since it has 2 rows
  and 2 columns. Also, since the number of rows
  and the number of columns are equal, we say that
  A is square.

3
Determinants

• Associated with every 2?2 matrix
  is a number called the
  determinant of A, denoted by A.
• A is defined by the rule
• Example.
• Although it is possible to discuss determinants
  for larger matrices, we limit ourselves to the
  2?2 case.

4
Summary of Determinants We discussed

• Definition of a matrix and its rows and columns
• Definition of the coefficient matrix of a system
• Determinant of a 2?2 matrix

5
Cramer's Rule

• Consider the general linear system of two
  equations in two unknowns
• Let A be the coefficient matrix of this system
• Let A1 denote the matrix obtained by replacing
  the first column of A with the column of the
  right-hand sides of the equations, and let A2 be
  defined in a similar fashion. Thus,

6
Cramer's Rule, continued.

• Cramer's Rule states The solution
  to is given by
• Problem. Solve by Cramer's Rule
  
  Solution.

7
Summary of Cramer's Rule We discussed

• The matrices A1 and A2 associated with a system
  of equations
• The statement of Cramer's Rule as x A1/A,
  y A2/A.