Q.Evaluate $\begin{vmatrix}0&1\\-1&0\end{vmatrix}$.

A determinant is a single number computed from a square array of numbers. For order 2, $\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc$. For order 3, expand along any row or column using cofactors: expanding along row 1,

Every determinant can be expanded along any of its 3 rows or 3 columns and gives the same value — a fact that follows from Property 1 (transpose invariance) of §4.2 and underlies why we're free to pick whichever row/column has the most zeros. The determinant of a matrix (as opposed to a bare grid of numbers) is defined only when the matrix is square — replace its square brackets with vertical bars to get $|A|=\det(A)$. If $|A|=0$, $A$ is called singular; otherwise non-singular. This single number packs in a huge amount of information: it tells you whether a linear system has a unique solution (Cramer's Rule, §4.3.1), whether three points are collinear (§4.3.3), and whether a matrix can be "undone" (has an inverse) — the last of these is developed further in the Class-12 continuation of this topic.

"Determinant of a matrix formula for order 2 and 3" and "determinants class 12 important questions" are among the most searched topics tied to the NCERT/CBSE Class 12 Mathematics Determinants chapter, since this concept underlies Cramer's-rule-style systems, collinearity checks and matrix invertibility questions in both board exams and JEE Main. Choosing a row or column with the most zeros before expanding, as noted here, is a genuine time-saving exam technique rather than just a textbook remark.