Determinant of a 2×2 Matrix

For $M = \begin{pmatrix}a & b\\ c & d\end{pmatrix}$, the determinant is $\det M = ad - bc$. So $\det\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix} = 1\cdot 4 - 2\cdot 3 = -2$. It measures the signed area of the parallelogram spanned by the columns.

Slide the four entries $a$, $b$, $c$, $d$. The lab returns $\det M = ad - bc$ live, and flags singularity when $\det M = 0$. Try $a = d = 1$, $b = c = 0$: $\det = 1$ (identity matrix).

When $\det M = 0$, the two column vectors are linearly dependent (one is a multiple of the other) and the parallelogram collapses to a line. The inverse formula $M^{-1} = \frac{1}{\det M}\begin{pmatrix}d & -b\\-c & a\end{pmatrix}$ divides by $\det M$, so a zero determinant means no inverse.

Solving a $2\times 2$ linear system uses Cramer's rule, which divides determinants. Checking if three points are collinear uses a $3\times 3$ determinant. In computer graphics, the determinant tells you whether a shape was flipped (reflection has $\det = -1$).