3.7

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3.7 Evaluate Determinants and Apply Cramers
Rule

• Associated with each square (n x n) matrix is a
  real number called its determinant. The
  determinant of a matrix A is denoted by det A or
  by A.
• The determinant of a 2 x 2 matrix is the
  difference of the products of the elements of the
  diagonals.

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3.7 Evaluate Determinants and Apply Cramers
Rule
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3.7 Evaluate Determinants and Apply Cramers
Rule

• Example 1
• Evaluate the determinant of the matrix.

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3.7 Evaluate Determinants and Apply Cramers
Rule

• Example 2
• Evaluate the determinant of the matrix.

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3.7 Evaluate Determinants and Apply Cramers
Rule

• You can use a determinant to find the area of a
  triangle whose vertices are points in a
  coordinate plane.

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3.7 Evaluate Determinants and Apply Cramers
Rule

• Example 3
• Off the coast of California lies a triangular
  region of the Pacific Ocean where huge
  populations of sea lions and seals live. The
  triangle is formed by imaginary lines connecting
  Bodega Bay, the Farallon Islands, and Ano Nuevo
  Island, as shown (next slide). Use the
  determinant to estimate the area of the region.

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3.7 Evaluate Determinants and Apply Cramers
Rule

• Example 3
• Units measured in miles.

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3.7 Evaluate Determinants and Apply Cramers
Rule

• You can use determinants to solve a system of
  linear equations. The method, called called
  Cramers rule and named after the Swiss
  mathematician Gabriel Cramer, uses the
  coefficient matrix of the linear system.

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3.7 Evaluate Determinants and Apply Cramers
Rule
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3.7 Evaluate Determinants and Apply Cramers
Rule

• Example 4
• Use Cramers rule to solve the system

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3.7 Evaluate Determinants and Apply Cramers
Rule
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3.7 Evaluate Determinants and Apply Cramers
Rule

• Example 5
• Use Cramers rule to solve the linear system.