Determinants

For $2\times 2$: $\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad-bc$. Main diagonal minus anti-diagonal.

For $n\times n$, expand along any row or column. Along row $i$: $\det(A) = \sum_{j=1}^{n}(-1)^{i+j}\,a_{ij}\,M_{ij}$, where the $(i,j)$ minor $M_{ij}$ is the determinant of the $(n-1)\times(n-1)$ submatrix obtained by deleting row $i$ and column $j$. The factor $(-1)^{i+j}$ is the cofactor sign.

Sign pattern: $(-1)^{i+j}$ creates a checkerboard starting with $+$ at $(1,1)$: $\begin{pmatrix}+&-&+\\-&+&-\\+&-&+\end{pmatrix}$. Always expand along the row or column with the most zeros to minimise computation.

For a triangular matrix (upper or lower), the determinant is simply the product of the diagonal entries. Row reducing to triangular form is often faster than cofactor expansion for large matrices.

Row operations have predictable effects: swapping two rows multiplies $\det$ by $-1$; scaling a row by $c$ multiplies $\det$ by $c$; adding a scalar multiple of one row to another leaves $\det$ unchanged. These three rules make row reduction a fast alternative to cofactor expansion.

Product rule: $\det(AB)=\det(A)\det(B)$. Composition scales volumes multiplicatively. Consequences: $\det(A^{-1})=1/\det(A)$ and $\det(A^k)=\det(A)^k$.

Transpose rule: $\det(A^T)=\det(A)$. Rows and columns play symmetric roles in the determinant.

Scaling: $\det(cA) = c^n\det(A)$ for an $n\times n$ matrix, because each of the $n$ rows is scaled by $c$.

A matrix has $\det=0$ whenever: two rows are equal, a row is all zeros, a row is a linear combination of the others (i.e., the rows are linearly dependent). All of these are equivalent to the columns being linearly dependent.

For an invertible system $A\mathbf{x}=\mathbf{b}$, Cramer's Rule gives each variable explicitly: $x_i = \det(A_i)/\det(A)$, where $A_i$ is the matrix $A$ with its $i$-th column replaced by $\mathbf{b}$.

Cramer's Rule is elegant but computationally expensive — computing $n+1$ determinants of $n\times n$ matrices is $O(n!)$ for each determinant via cofactor expansion, far slower than $O(n^3)$ Gaussian elimination.

Its primary value is theoretical. The formula shows that solutions are rational functions of the entries of $A$ and $\mathbf{b}$, which is useful in sensitivity analysis and proofs involving matrix calculus (e.g., the derivative of the inverse).

For a $2\times 2$ matrix $A$, $|\det(A)|$ equals the area of the parallelogram spanned by the columns (or rows) of $A$. For $3\times 3$, it equals the volume of the parallelepiped.

If $\det(A)>0$, the transformation preserves orientation (counterclockwise goes to counterclockwise). If $\det(A)<0$, orientation is reversed (a reflection is involved).

If $\det(A)=0$, the columns are linearly dependent and the transformation collapses the full-dimensional space to a lower-dimensional one — it maps $\mathbb{R}^n$ onto a proper subspace.

The determinant is multilinear and alternating in the columns: scaling one column by $c$ scales $\det$ by $c$; swapping two columns flips the sign. The parallelogram area interpretation makes this geometric: scaling a side scales the area; reflecting a side flips orientation.