Determinant

For a pair of simultaneous linear equations $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$, you already know the system has a unique solution precisely when $a_1b_2 - a_2b_1 \neq 0$. That single expression, $a_1b_2 - a_2b_1$, built from the coefficient matrix of the system, turns out to be important enough to deserve its own name and notation: it is called the determinant of the coefficient matrix.

More generally, every square matrix $A$ has associated with it a single number (real or complex), called its determinant, written $\det A$ or $|A|$ (sometimes using the Greek letter $\Delta$). Formally, a determinant is a function mapping a square matrix to a number: $f : M \to K$, where $M$ is the set of square matrices of a given order and $K$ is the set of numbers.