Determinant

Formula#

Closed forms for small $n$ (MML (4.5)–(4.8)):

• $n = 2$: $\det\begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = a_{11}a_{22} - a_{12}a_{21}$.
• $n = 3$ (Sarrus’ rule): $\det(A) = a_{11}a_{22}a_{33} + a_{21}a_{32}a_{13} + a_{31}a_{12}a_{23} - a_{31}a_{22}a_{13} - a_{11}a_{32}a_{23} - a_{21}a_{12}a_{33}$.
• Triangular $T$ (all entries below, or all above, the diagonal are $0$): $\det(T) = \prod_{i=1}^n T_{ii}$.

Properties#

See Invertible Matrix > Properties for the other equivalent conditions. Geometrically, $\det(A) = 0$ means the columns are linearly dependent, so they span a flat parallelepiped with volume $0$.

Signed volume (MML Example 4.2): $|\det(A)|$ is the volume of the parallelepiped spanned by the columns $a_1, \dots, a_n$ of $A$, and for $n = 2$ the area of the parallelogram with vertices $0, a_1, a_2, a_1 + a_2$. The sign gives the orientation of $(a_1, \dots, a_n)$ relative to the standard basis $(e_1, \dots, e_n)$. Swapping two columns flips it.

Rules for $A, B \in \mathbb{R}^{n \times n}$ (MML p. 103):

• $\det(AB) = \det(A)\det(B)$.
• $\det(A^\top) = \det(A)$. So every statement about columns holds for rows as well.
• $\det(A^{-1}) = \dfrac{1}{\det(A)}$ if $A$ is invertible (from $\det(A)\det(A^{-1}) = \det(I) = 1$).
• Similar matrices $B = S^{-1}AS$ have the same determinant. So $\det$ is a property of the linear map, independent of the basis.
• Adding a multiple of one row/column to another doesn’t change $\det(A)$.
• Multiplying one row/column by $\lambda \in \mathbb{R}$ multiplies $\det(A)$ by $\lambda$. In particular $\det(\lambda A) = \lambda^n \det(A)$.
• Swapping two rows/columns changes the sign of $\det(A)$.
• $\det(A) = \prod_{i=1}^n \lambda_i$, the product of the eigenvalues counted with algebraic multiplicity (MML Thm. 4.16).

Computing it: the last three row rules say how each step of Gaussian elimination changes $\det$. Reduce $A$ to triangular form, multiply the diagonal entries, and correct the sign once per row swap. This costs $O(n^3)$ operations.

Characteristic polynomial: $p_A(\lambda) = \det(A - \lambda I)$ is a polynomial in $\lambda$ whose roots are the eigenvalues of $A$ (see Eigenvalue > Characteristic polynomial).

Example#

Intuition#

A matrix $A$ sends $e_1$ to its first column $a_1$ and $e_2$ to its second column $a_2$, so the unit square becomes the parallelogram spanned by $a_1, a_2$. Its area $|\det(A)|$ is the factor by which $A$ scales every area: the image of the grid consists of equal parallelograms, each $|\det(A)|$ times as large as a grid cell, and any shape can be filled with ever finer grid cells. So the disk of area $0.79$ becomes an ellipse of area $2.5 \cdot 0.79 = 1.96$. So $\det$ is not a property of the square but of the map.

The sign tells whether $A$ keeps or flips orientation. Turning from $e_1$ to $e_2$ goes counterclockwise. If the turn from $a_1$ to $a_2$ does too, $\det(A) > 0$ and the letter F stays readable. If the turn is clockwise (here: the columns swapped), $\det(A) < 0$ and the F comes out mirrored. In between lies $\det(A) = 0$: the columns are parallel ($a_2 = \frac12 a_1$), the whole plane is squashed onto a line, and that information can’t be undone. This is why $\det(A) = 0$ means not invertible.

Applying $B$ scales areas by $\det(B)$, and applying $A$ afterwards scales the result by $\det(A)$. Together the area changes by $\det(A)\det(B)$, which gives $\det(AB) = \det(A)\det(B)$ without any computation. The same picture gives $\det(A^{-1}) = 1/\det(A)$, because $A^{-1}$ must undo the scaling.

Sources#

• MML pp. 99–100 (motivation, $2 \times 2$ case, Thm. 4.1), MML p. 101 (triangular matrices, Example 4.2: signed volume), MML pp. 102–103 (Laplace expansion Thm. 4.2, Example 4.3, rules, Thm. 4.3)
• MML p. 104 (characteristic polynomial, $c_0 = \det(A)$), MML p. 112 (Thm. 4.16: $\det$ = product of eigenvalues)
• Use in MSI: normalization $1/\sqrt{\det(2\pi\Sigma)}$ of the Multivariate Normal Distribution; invertibility tests for $\Phi^\top\Phi$ and Jacobians (Invertible Matrix, Jacobian Matrix)