Final answer:
The identity matrix is invertible because it serves as its own inverse. Distances are invariant under rotations of the coordinate system because the Pythagorean theorem remains valid. The sum of light intensities through polarizing filters at angles θ and 90° equals the original intensity due to the trigonometric identity cos2(θ) + sin2(θ) = 1.
Step-by-step explanation:
To prove that the identity matrix is invertible, we need to consider the definition of an invertible matrix. A matrix is considered invertible if there exists another matrix that when multiplied with it, results in the identity matrix. For the identity matrix, denoted as I, if you multiply it by itself (I * I), the result is still the identity matrix. Therefore, the identity matrix serves as its own inverse, proving it is invertible.
Invariance Under Coordinate Rotation
For part (b) and (c), when dealing with rotations of the coordinate system, the distance from any point to the origin, as well as the distance between two points, remains the same after rotation. The equations representing these distances are derived from the Pythagorean theorem, which remains true regardless of coordinate rotation. Hence, distances are invariant under rotations.
Intensity of Light Through Polarizing Filters
In problem 90, concerning the intensity of light transmitted through polarizing filters, we use the fact that cos2(θ) + sin2(θ) = 1 to show that I + I' = Io. Here, Io represents the original intensity, and I and I' represent the intensities after light passes through the filter at angles θ and 90°, respectively. Since cos(90°) = sin(θ) for any angle θ, it follows that the sum of the intensities after the two filters is equal to the original intensity.