Determinants

At a glance 11 formulas

1. 12×2 determinant$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$
2. 23×3 expansionExpand along the row or column with the most zeros: each entry times the 2×2 left when its row and column are deleted, with checkerboard signs $+\,-\,+$ from the top left.
3. 3Triangular matrixWhen every entry below (or above) the diagonal is 0, the determinant is the product of the diagonal entries, not their sum.
4. 4Determinant is zeroTwo equal or proportional rows (or columns), or a row of zeros, make the determinant 0. Look for them before expanding.
5. 5Row operations$R_i \leftrightarrow R_j:\ \text{sign flips}$$R_i \to kR_i:\ \times k;\ R_i \to R_i + kR_j:\ \text{no change}$
6. 6Scalar multiple$\det(kA) = k^n \det A$$n = \text{order of } A$
7. 7Product, transpose, inverse$\det(AB) = \det A \cdot \det B$$\det A^T = \det A,\ \det A^{-1} = \dfrac{1}{\det A}$
8. 8Adjoint$\det(\text{adj}\,A) = (\det A)^{n-1}$$A\,(\text{adj}\,A) = (\det A)\,I,\ n = \text{order},\ I = \text{identity}$
9. 9Singular matrix$\det A = 0 \iff A^{-1} \text{ does not exist}$$\det(A^2 + kA) = \det A \cdot \det(A + kI),\ I = \text{identity}$
10. 10Rows 1, a, a²$\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix} = (b - a)(c - a)(c - b)$
11. 11Cyclic rows$\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = 3abc - a^3 - b^3 - c^3$$= -(a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$