Summary

This chapter built the theory of determinants for square matrices up to order $3\times3$ and used them to solve linear systems. Starting from the direct formula $|A|=a_{11}a_{22}-a_{12}a_{21}$ for a $2\times2$ matrix, the $3\times3$ determinant was defined by expansion along the first row, and then generalised using minors $M_{ij}$ and cofactors $C_{ij}=(-1)^{i+j}M_{ij}$ to justify expansion along any row or column. Six key properties of determinants -- transpose-invariance, sign-reversal on row/column swap, the zero value for identical rows/columns, factoring out a common multiplier, the sum property, and the row/column-combination invariance -- were introduced as tools for simplifying a determinant before expanding it. These ideas gave the compact area-of-a-triangle formula $\tfrac12\left|\begin{smallmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{smallmatrix}\right|$ and its collinearity corollary (area $=0$). The adjoint of $A$ -- the transpose of its cofactor matrix -- satisfies the key relation $A(\operatorname{adj}A)=|A|I$, from which the inverse formula $A^{-1}=\tfrac{1}{|A|}\operatorname{adj}A$ follows directly whenever $A$ is non-singular ($|A|\ …