Determinant (legacy)

1. Fundamental Properties

• Multilinearity in rows (or columns): linear in each row separately.
• Alternating: if two rows are equal, $\det =0$.
• Normalization: $\det (I_n)=1$.
• Product rule:

$$\det (AB)=\det (A)\det (B).$$

• Transpose invariance:

$$\det (A^T)=\det (A).$$

• Invertibility criterion:

$$A\text{ is invertible}⟺\det (A)\ne 0.$$

2. Equivalent Statements to $\det (A)\ne 0$

• Invertibility $A$ is invertible (i.e. there exists $A^{-1}$).

• Full rank $\mathrm{rank}(A)=n$.

• Trivial kernel $\ker (A)=\{0\}$.

• Homogeneous system The only solution of $Ax=0$ is the trivial one, $x=0$.

• Unique solvability For every $b\in {\mathbb{R}}^n$, the equation $Ax=b$ has exactly one solution.

• Linear independence of columns The columns of $A$ form a basis of ${\mathbb{R}}^n$.

• Linear Independence of rows The rows of $A$ form a basis of ${\mathbb{R}}^n$.

• Bijectivity of the associated map The linear map $T:{\mathbb{R}}^n\to {\mathbb{R}}^n,T(x)=Ax$ is bijective.

• Eigenvalue condition $0$ is not an eigenvalue of $A$.

• Characteristic polynomial If $p_A(\lambda )=\det (\lambda I-A)$, then $p_A(0)\ne 0$.

• Minimal polynomial The minimal polynomial of $A$ does not have $0$ as a root.

• Adjugate formula $\mathrm{adj}(A)$ exists and $A^{-1}=\frac{1}{\det A}\mathrm{adj}(A).$

• LU-decomposition without pivoting $A$ admits an $LU$ factorization with no row exchanges.

• Volume interpretation As the linear map’s oriented-volume scale factor, $\det A$ is nonzero iff the map is volume–nondegenerate.

3. Computation Methods

• Laplace (cofactor) expansion along any row or column.
• Row-reduction to upper triangular form:

$$\det (A)=(\text{product of pivots})\times (-1{)}^{\#\text{swaps}}.$$

• LU decomposition: if $A=LU$ with unit-lower $L$, then $\det (A)=\prod \mathrm{diag}(U)$.

4. Cofactor & Adjugate

• Cofactor $C_{ij}=(-1{)}^{i+j}\det (M_{ij})$.
• Adjugate $\mathrm{adj}(A)=[C_{ji}]$.
• Inverse formula (when $\det (A)\ne 0$):

$$A^{-1}=\frac{1}{\det (A)}\mathrm{adj}(A).$$

5. Cramer’s Rule

For $A\mathbf{x}=\mathbf{b}$,

$$x_i=\frac{\det (A_i)}{\det (A)},$$

where $A_i$ replaces the $i$th column of $A$ with $\mathbf{b}$.

6. Effect of Elementary Operations

• Row swap: flips sign, $\det \mapsto -\det$.
• Scaling a row by $\alpha$: multiplies $\det$ by $\alpha$.
• Adding a multiple of one row to another: leaves $\det$ unchanged.

7. Geometric Interpretation

• $\det (A)$ is the oriented volume scaling factor of the linear map $A:{\mathbb{R}}^n\to {\mathbb{R}}^n$.
• $\det (A)>0$ preserves orientation; $\det (A)<0$ reverses it.