Why does a numerical linear algebra algorithm, purportedly proving matrix invertibility, fail for ill-conditioned matrices?

💡 Explanation

A supposedly-valid proof of invertibility may fail due to the algorithm being sensitive to round-off errors. Ill-conditioned matrices amplify these errors due to high condition number, therefore, the calculated inverse might not be accurate, rather than inherent limitations of matrix invertibility proofs themselves.

• A sample of Potassium-40 undergoes beta decay. Which outcome increases given the constant decay energy release?
• An engineer designs a solid-oxide fuel cell (SOFC) stack operating at 800°C. Which risk increases as the operational enthalpy deviates negatively from the design value?
• If an antenna is designed using theoretical electrodynamics to emit a radio wave at a specific frequency, which consequence follows from applying the Maxwell-Lorentz equations?
• A ruby laser's flash lamp intensity is increased, yet the laser output remains unchanged. Which mechanism explains this saturation?
• What causes variations in the thermal conductivity of a crystalline solid when comparing different crystallographic directions?
• What causes differences when calculating the theoretical wave equation using Maxwell's equations versus the paraxial approximation in optics?