Cramer’s rule, inverse matrix, and volume
Problem 20.1: (5.3 #8. Introduction to Linear Algebra: Strang) Suppose
Find its cofactor matrix C and multiply ACT to find det(A).
C and ACT = .
If you change a1,3 = 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.”