4 Determinants

To every square matrix this section attaches a single number which decides whether the matrix can be inverted. That such a number exists at all is not obvious.

The determinant is built up by size — \(2\) by \(2\), then \(3\) by \(3\), then \(n\) by \(n\) — each defined recursively in terms of the one below by expansion along a row. The properties established along the way are what make it usable: it changes sign when two rows are exchanged, vanishes when two rows are equal, and is multiplicative, \(\det (AB)=\det A\det B\).

Two formulae close the section. The adjoint gives \(A^{-1}\) explicitly, and Cramer’s rule gives the solution of a square system as a ratio of determinants. Neither is an efficient way to compute anything, and the section says so; their value is that they express the answer as a formula in the entries, which is what later arguments need.

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