[Solved] 18.06 Unit 2 Exercise 7-Cramer’s rule, inverse matrix, and volume

Cramer’s rule, inverse matrix, and volume

Problem 20.1: (5.3 #8. Introduction to Linear Algebra: Strang) Suppose

A .

Find its cofactor matrix C and multiply AC^T to find det(A).

C and AC^T = .

If you change a_1,3 = 4 to 100, why is det(A) unchanged?

Problem 20.2: (5.3 #28.) Spherical coordinates ρ, φ, θ satisfy

x = ρ sin φ cos θ, y = ρ sin φ sin θ and z = ρ cos φ.

Find the three by three matrix of partial derivatives:

.

Simplify its determinant to J = ρ² sin φ. In spherical coordinates,

dV = ρ² sin φ dρ dφ dθ

is the volume of an infinitesimal “coordinate box.”

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