Someone argues: "The matrix A = pmatrix 0.01 & 0 0 & 0.01 pmatrix has

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Numerical and computational mathematics

Conditions of Validity, Ch The four subspaces and the Fredholm alternative

Problem

Someone argues: "The matrix A=(0.01000.01)\mathsf{A} = \begin{pmatrix} 0.01 & 0 \\ 0 & 0.01 \end{pmatrix} has det⁡A=1.0×10−4\det\mathsf{A} = 1.0\times10^{-4}, which is tiny. A tiny determinant means A\mathsf{A} is nearly singular, so solving Ax=b\mathsf{A}\mathbf{x} = \mathbf{b} for it must amplify any error in b\mathbf{b} enormously." ((a)) Find the error in this reasoning by computing κ(A)\kappa(\mathsf{A}). ((b)) State, in one sentence, what the smallness of det⁡A\det\mathsf{A} is actually telling you here.

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