The Determinant of a 2×2 Matrix

This WBCHSE Class 12 Commerce Business Mathematics and Basic Statistics chapter builds directly on the previous chapter's matrices to introduce the determinant — a single number computed from a square matrix — and then uses it to solve a system of two linear equations in two unknowns by Cramer's Rule, a systematic alternative to the elimination/substitution methods.

What is a determinant?

For a $2\times2$ matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, the determinant of $A$, written $|A|$ or $\det A$ or $\begin{vmatrix}a&b\\c&d\end{vmatrix}$, is the single number

the product of the leading-diagonal entries minus the product of the other-diagonal entries. For example, for $A=\begin{pmatrix}4&3\\2&5\end{pmatrix}$,

Determinant of the identity matrix

For the $2\times2$ identity matrix $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}$, $|I|=1(1)-0(0)=1$. A determinant of exactly $0$ signals something special about a matrix (its rows/columns are proportional to each other) — this chapter's Cramer's Rule application (Section 2) depends heavily on whether a particular determinant is zero or not.

The determinant of a 2×2 matrix is a standard tool across commerce-mathematics and algebra curricula nationally, and is the foundation this WBCHSE syllabus builds on to solve linear systems by Cramer's Rule, next.