Determinants of 2x2 and 3x3 Matrices

For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ the determinant is $ad - bc$: down-product minus up-product.

For a $3 \times 3$, expand along whichever row or column carries the most zeros. Along the top row, $|A| = a(ei - fh) - b(di - fg) + c(dh - eg)$ — note the alternating sign.

If any row is a multiple of another, or a row is all zeros, the determinant is $0$ and there is nothing to compute. Check that first.