Math Problem Solver

Certainly, let's calculate the determinant of the Jacobian matrix of g at f(2,1).

1. Calculate f(2, 1):

f(2, 1) = (2^2 - 1^2, 2 * 2 * 1) = (3, 4)

2. Calculate the Jacobian Matrix of f(x, y):

The Jacobian matrix of f(x, y) is given by:

J_f(x, y) = | ∂f₁/∂x ∂f₁/∂y |

| ∂f₂/∂x ∂f₂/∂y |

where f₁ = x² - y² and f₂ = 2xy

J_f(x, y) = | 2x -2y |

| 2y 2x |

3. Calculate the Jacobian Matrix of f at (2, 1):

J_f(2, 1) = | 4 -2 |

| 2 4 |

4. Calculate the Determinant of J_f(2, 1):

det(J_f(2, 1)) = (4 * 4) - (-2 * 2) = 16 + 4 = 20

5. Calculate the Inverse of J_f(2, 1):

The inverse of J_f(2, 1) is given by:

J_f(2, 1)^-1 = 1/det(J_f(2, 1)) * | 4 2 |

| -2 4 |

J_f(2, 1)^-1 = 1/20 * | 4 2 |

| -2 4 |

J_f(2, 1)^-1 = | 1/5 1/10 |

| -1/10 1/5 |

6. Calculate the Determinant of J_f(2, 1)^-1:

det(J_f(2, 1)^-1) = (1/5 * 1/5) - (-1/10 * 1/10) = 1/25 + 1/100 = 0.05

Therefore, the determinant of the Jacobian matrix of g at f(2, 1) is 0.05 (rounded to two decimal places).