Inverse and conditioning

Undoing a map

A square matrix $A$ has an inverse $A^{-1}$, with $A^{-1}A=AA^{-1}=I$, exactly when its map loses no information: different inputs give different outputs. The determinant measures how $A$ scales volumes (with a sign for flips), so $A$ is invertible exactly when $\det A\neq0$. For a $2\times2$ matrix, $A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$. Useful rules: $\det(AB)=\det A\,\det B$ and $\det(A^{-1})=1/\det A$.

Invertible is not the same as safe. The condition number $\kappa(A)$ says how much a small relative error in $\mathbf{b}$ (or in the entries of $A$) can be magnified in the solution $\mathbf{x}$. A rule of thumb: you lose about $\log_{10}\kappa$ of your roughly 16 digits of double precision. A small determinant alone proves nothing, because it depends on the scale: $0.001\,I$ has a tiny determinant but $\kappa=1$. Nearly collinear columns are what make $\kappa$ huge.

• Invertible: square with $\det A\neq0$; no information is lost.
• Condition number: how much errors in the data are magnified; large means a fragile solve.
• Never form $A^{-1}$ to solve: a factorisation solve is faster and more accurate.

Lab · what's graded

• The inverse of a 2 by 2 matrix20 pts
• A 3 by 3 determinant10 pts
• Find the singular matrix15 pts
• Digits you can trust15 pts
• Good numerical practice15 pts
• Determinant rules10 pts
• Why not invert15 pts