Properties of Determinants

Q: Is the determinant unchanged by a change of basis?

Yes. A change of basis replaces $A$ by $P^{-1}AP$, and multiplicativity gives $\det(P)^{-1}\det(A)\det(P) = \det(A)$. The determinant therefore belongs to the underlying linear map rather than to any particular matrix representation of it, which is why it can be used to define the determinant of an operator.

Overview

The determinant earns its place not through the recursive definition, which is unwieldy, but through a small set of structural properties. Those properties let a determinant be reasoned about, transformed and combined without ever being expanded. They are also what connect the determinant to the rest of linear algebra: to invertibility, to rank, to eigenvalues and to the geometry of linear maps.

Two of the properties describe how the determinant interacts with matrix operations. Transposition leaves it entirely alone, which immediately explains why expansion about a row and expansion about a column give the same answer. Multiplication behaves as well as it possibly could: the determinant of a product is the product of the determinants. That multiplicativity is genuinely surprising — matrix multiplication is not commutative, involves every entry of both matrices, and has no obvious reason to respect a single scalar summary — and it is the property that most of the useful corollaries flow from.

The third property is the singularity criterion. A square matrix is singular precisely when its determinant is zero. This converts a structural question about null spaces, rank and solution uniqueness into a single arithmetic test, and it slots directly into the running list of conditions equivalent to non-singularity. For a matrix containing a design parameter it does more: setting the determinant to zero and solving for the parameter locates every value at which the modelled system degenerates.

One property that does not hold is worth stating as loudly as the ones that do. The determinant is not additive: $\det(A + B)$ bears no simple relationship to $\det(A)$ and $\det(B)$ , and assuming otherwise is a common and consequential error. The determinant is a highly non-linear function of the matrix entries — it is linear in each row separately, but not in the matrix as a whole.

Definition

Determinant of the Transpose

For any square matrix $A$ , $\det(A^{t}) = \det(A)$ . Transposing a matrix reflects it across the main diagonal, exchanging rows for columns, and leaves the determinant unaltered.

The proof is an induction on size. Expanding $\det(A)$ about row $1$ and $\det(A^{t})$ about column $1$ pairs the terms exactly, because $[A^{t}]_{j1} = [A]_{1j}$ and the sign $(-1)^{i+j}$ is symmetric in its indices; the induction hypothesis identifies the corresponding cofactors.

Determinant Respects Matrix Multiplication

DRMM

For square matrices $A$ and $B$ of the same size, $\det(AB) = \det(A)\det(B)$ . The determinant of a product is the product of the determinants, in either order, notwithstanding that $A B$ and $B A$ are generally different matrices.

Singular Matrices have Zero Determinants

SMZD

A square matrix $A$ is singular if and only if $\det(A) = 0$ . Equivalently, $A$ is non-singular if and only if $\det(A) \ne 0$ . Recall that $A$ is singular when the homogeneous system $A\vec{x} = \vec{0}$ has a solution other than $\vec{x} = \vec{0}$ .

The mechanism is row reduction. Each elementary row operation multiplies the determinant by a non-zero factor — by $1$ for adding a multiple of one row to another, by $-1$ for an interchange, by $\alpha \ne 0$ for a scaling — so the determinant of a matrix is zero exactly when the determinant of its reduced row-echelon form is. A square matrix in reduced row-echelon form is either the identity, with determinant $1$ , or has a zero row, with determinant $0$ .

Concepts

Why transpose invariance matters more than it looks

Transpose invariance is the reason the determinant has no preferred orientation. Every statement about rows transfers verbatim to columns: expansion about a column is legitimate because it is expansion about a row of the transpose; a matrix with two equal columns has zero determinant because its transpose has two equal rows; the column operations act on the determinant exactly as the row operations do. Rather than proving each column statement separately, one proves the row statement and transposes.

Multiplicativity and its immediate corollaries

From $\det(AB) = \det(A)\det(B)$ a chain of results follows with almost no further work. Taking $B = A^{-1}$ and using $\det(I_n) = 1$ gives $\det(A^{-1}) = 1/\det(A)$ , which also re-proves that a singular matrix cannot have an inverse. Iterating gives $\det(A^{k}) = \left(\det(A)\right)^{k}$ . Writing $\alpha A = (\alpha I_n)A$ and observing that $\alpha I_n$ is diagonal with determinant $\alpha^{n}$ gives $\det(\alpha A) = \alpha^{n}\det(A)$ — the $n$ -th power being the reason determinant magnitude is a poor conditioning measure. Similar matrices $B = P^{-1}AP$ satisfy $\det(B) = \det(P)^{-1}\det(A)\det(P) = \det(A)$ , so the determinant is a similarity invariant.

The determinant is not additive

There is no useful identity relating $\det(A + B)$ to $\det(A)$ and $\det(B)$ . A two-line counterexample settles it: with $A = I_2$ and $B = -I_2$ , $\det(A) = \det(B) = 1$ while $\det(A + B) = \det(O) = 0$ . What is true is a weaker statement: the determinant is a linear function of any one row when all the other rows are held fixed. That row-wise multilinearity, together with the alternating property, actually characterises the determinant uniquely up to normalisation, but it is emphatically not linearity in the matrix.

The singularity criterion as an equivalence

Because $A$ is singular if and only if $\det(A) = 0$ , negating both sides gives an equally usable statement: $A$ is non-singular if and only if $\det(A) \ne 0$ . This joins a long list of mutually equivalent characterisations of non-singularity, so a single determinant evaluation certifies simultaneously that $A$ row-reduces to the identity, that $\mathcal{N}(A) = \{\vec{0}\}$ , that $A\vec{x} = \vec{b}$ has a unique solution for every $\vec{b}$ , that the columns of $A$ are linearly independent and form a basis of $\mathbb{C}^n$ , that $A$ is invertible, that $\mathcal{C}(A) = \mathbb{C}^n$ , that $\operatorname{rank}(A) = n$ and that $\operatorname{nullity}(A) = 0$ .

Determinant versus row reduction as a singularity test

Conceptually the determinant looks like the more efficient test: it uses only addition, subtraction and multiplication, so no division-induced rounding arises, and the final question — zero or not — is trivial. Computationally the position reverses. Evaluating a determinant from the definition costs more than $n!$ operations, whereas row reduction settles singularity in about $\tfrac{2}{3}n^3$ . In floating point the determinant is worse still, because rounding turns an exact zero into a small non-zero value and the answer must be judged against a threshold that the determinant's scaling behaviour makes almost impossible to set.

Parametric determinants and critical values

The highest-value engineering use of these properties is symbolic. Build a matrix whose entries contain a design parameter, expand the determinant into a polynomial in that parameter, and set it to zero. The roots are exactly the parameter values at which the modelled system becomes singular: the buckling loads of a structure, the poles of a transfer function, the eigenvalues of a matrix, the component values at which a network loses a unique solution. Because the determinant requires no division, the resulting polynomial has coefficients that are polynomial in the original entries and can be factored exactly.

Using determinant properties to answer a question about a matrix

Is the matrix a product?If

M = AB

M = P^{-1}AP

, apply multiplicativity rather than forming the product. A similarity transform leaves the determinant unchanged.

Is it a transpose or an adjoint?

\det(A^{t}) = \det(A)

exactly; for the conjugate transpose,

\det(A^{*}) = \overline{\det(A)}

Does it contain a parameter?Expand symbolically into a polynomial and factor. The roots are the parameter values at which the matrix becomes singular.

Is a numeric singularity decision needed?In exact arithmetic, test

\det(A) = 0

directly. In floating point, use rank or the condition number instead — determinant magnitude is not a reliable proxy.

Confirm the structural consequencesA non-zero determinant certifies invertibility, full rank, a trivial null space and a unique solution for every right-hand side, all at once.

Equations

Determinant of the transpose

EQ-PDM-01

\det (A^{t}) = \det (A)

The determinant is blind to the row-column distinction, which is why expansion about a column is as valid as expansion about a row.

Determinant respects matrix multiplication

EQ-PDM-02

\det (A B) = \det (A) \det (B)

For square matrices of the same size. Since the right-hand side is a product of scalars it is symmetric, so $\det(AB) = \det(BA)$ even though $AB \ne BA$ in general.

Singularity criterion

EQ-PDM-03

A singular \Leftrightarrow \det (A) = 0, A non-singular \Leftrightarrow \det (A) \neq 0

The equivalence that connects the determinant to every other characterisation of non-singularity.

Determinant of an inverse

EQ-PDM-04

\det (A^{- 1}) = \frac{1}{\det (A)}, \det (I_{n}) = 1

Immediate from multiplicativity applied to $AA^{-1} = I_n$ . It also shows that a matrix with zero determinant cannot possibly have an inverse.

Scalar multiplication and powers

EQ-PDM-05

\det (α A) = α^{n} \det (A), \det (A^{k}) = (\det (A))^{k}

The $n$ -th power in the first identity is why the magnitude of a determinant says nothing about conditioning: rescaling a matrix rescales its determinant enormously.

Similarity invariance

EQ-PDM-06

B = P^{- 1} A P \Rightarrow \det (B) = \det (P)^{- 1} \det (A) \det (P) = \det (A)

The determinant depends only on the underlying linear map, not on the basis used to represent it. This is what makes it a legitimate invariant of an operator.

Failure of additivity

EQ-PDM-07

\det (A + B) \neq \det (A) + \det (B) in general

Take $A = I_2$ and $B = -I_2$ : both determinants are $1$ , but $A + B$ is the zero matrix with determinant $0$ . The determinant is multilinear in the rows, not linear in the matrix.

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$A$	Matrix	The square matrix under discussion	n x n matrix over C
$B$	Second matrix	A square matrix of the same size, used in product and sum identities	n x n matrix over C
$n$	Size	Common row and column count of the square matrices	positive integer
$A^{t}$	Transpose	The matrix with rows and columns exchanged, $[A^{t}]_{ij} = [A]_{ji}$	n x n matrix
$A^{- 1}$	Inverse	The matrix with $AA^{-1} = A^{-1}A = I_n$ , existing exactly when $\det(A) \ne 0$	n x n matrix
$I_{n}$	Identity matrix	The size- $n$ identity, with $\det(I_n) = 1$	n x n matrix
$α$	Scalar	A scalar multiplier applied to a whole matrix or to one row	complex scalar
$N (A)$	Null space	Solutions of $A\vec{x} = \vec{0}$ ; trivial exactly when $\det(A) \ne 0$	subspace of C^n
$k$	Design parameter	A tunable entry appearing symbolically in the matrix	real or complex scalar

Worked Numerical Example

Problem statement

A three-node coupled network is modelled by a coefficient matrix containing one tunable coupling term $k$ . Determine the value of $k$ at which the model becomes singular, confirm the degeneracy directly, and then exercise the transpose and product properties on a non-singular instance of the same matrix.

Write the parametric matrix
The parameter $k$ occupies the $(3,3)$ position; every other entry is fixed by the network topology. The matrix is deliberately non-symmetric so that the transpose property has something to demonstrate.
$A (k) = [\begin{matrix} 2 & - 1 & 0 \\ - 3 & 2 & - 1 \\ 0 & - 1 & k \end{matrix}]$
Expand the determinant symbolically
Expand about the first row. The third term vanishes because $[A]_{13} = 0$ , leaving two $2 \times 2$ determinants: $\begin{vmatrix} 2 & -1 \\ -1 & k \end{vmatrix} = 2k - 1$ and $\begin{vmatrix} -3 & -1 \\ 0 & k \end{vmatrix} = -3k$ .
$\det (A (k)) = 2 (2 k - 1) - (- 1) (- 3 k) + 0 = 4 k - 2 - 3 k = k - 2$
Apply the singularity criterion
The determinant is a first-degree polynomial in $k$ with a single root. By the singularity criterion, $A(k)$ is singular for exactly one value of the coupling parameter and non-singular for every other value.
$\det (A (k)) = 0 \Leftrightarrow k = 2$
Confirm the degeneracy directly
Singularity means a non-trivial null vector exists. At $k = 2$ the vector below satisfies $A(2)\vec{x} = \vec{0}$ : the rows evaluate to $2 - 2 + 0 = 0$ , $-3 + 4 - 1 = 0$ and $0 - 2 + 2 = 0$ . This is an independent confirmation that does not reuse the determinant arithmetic.
$A (2) \vec{x} = [\begin{matrix} 2 & - 1 & 0 \\ - 3 & 2 & - 1 \\ 0 & - 1 & 2 \end{matrix}] [\begin{matrix} 1 \\ 2 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$
Fix a non-singular instance and check transpose invariance
Set $k = 5$ , giving $\det(A) = 3$ . Transposing produces a genuinely different matrix, since $A$ is not symmetric, yet the determinant is unchanged: expanding $A^{t}$ about its first row gives $2(2 \cdot 5 - (-1)(-1)) - (-3)((-1)(5) - (-1)(0)) + 0 = 2(9) + 3(-5) = 3$ .
$Fix a non-singular instance and check transpose invariance$
Form a product and test multiplicativity
Take a second matrix $B$ with $\det(B) = -4$ , obtained by expanding about its second row: $\det(B) = 1 \cdot \begin{vmatrix} 1 & 2 \\ 3 & 2 \end{vmatrix} = 2 - 6 = -4$ . Multiplicativity predicts $\det(AB) = 3 \times (-4) = -12$ without forming $A B$ at all.
$B = [\begin{matrix} 1 & 0 & 2 \\ 0 & 1 & 0 \\ 3 & 1 & 2 \end{matrix}], A B = [\begin{matrix} 2 & - 1 & 4 \\ - 6 & 1 & - 8 \\ 15 & 4 & 10 \end{matrix}]$
Verify the prediction by direct expansion
Expanding $\det(AB)$ about its first row gives $2\bigl((1)(10)-(-8)(4)\bigr) + 1\bigl((-6)(10)-(-8)(15)\bigr) + 4\bigl((-6)(4)-(1)(15)\bigr)$ . The three brackets are $42$ , $60$ and $-39$ .
$\det (A B) = 2 (42) + 1 (60) + 4 (- 39) = 84 + 60 - 156 = - 12$
Show that additivity fails
The same two matrices settle the additive question. $A + B$ has determinant $27$ , whereas $\det(A) + \det(B) = 3 - 4 = -1$ . The two numbers are unrelated, which is the general situation.
$A + B = [\begin{matrix} 3 & - 1 & 2 \\ - 3 & 3 & - 1 \\ 3 & 0 & 7 \end{matrix}], \det (A + B) = 27 \neq 3 + (- 4)$

Result

The network model degenerates at exactly one coupling value, $k = 2$ , where the coefficient matrix acquires the null direction $(1, 2, 1)$ — a state pattern that the model cannot distinguish from zero, so no unique solution exists there. Away from that value the matrix is non-singular and invertible. The transpose and product checks confirm both properties numerically, while the failed additive check demonstrates that determinants may be combined across matrix products but never across matrix sums.

Applications & Industry Use

Structural engineering

Critical buckling loads

Linear buckling analysis assembles an elastic stiffness matrix and a geometric stiffness matrix and seeks the load factor $\lambda$ at which $\det(K_{e} + \lambda K_{g}) = 0$ . Each root is a critical load and the corresponding null vector is a buckling mode shape. The singularity criterion is what makes a determinant condition equivalent to the existence of a deformation requiring no additional load.

Control systems

Poles, stability and the return difference

The characteristic equation $\det(sI - A) = 0$ locates the poles of a state-space model. In multivariable feedback design the return difference determinant $\det(I + L(s))$ plays the same role for the closed loop, and multiplicativity is what allows loop transfer functions in cascade to have their determinants multiplied rather than their matrices composed.

Mechanics and kinematics

Jacobian singularities of a manipulator

A robotic arm loses a degree of freedom at configurations where the Jacobian determinant vanishes. Because the determinant is a similarity invariant, the singular configurations do not depend on the coordinate frame chosen to express the kinematics, which is essential for a result that must hold regardless of how the model was parameterised.

Statistics and estimation

Log-likelihood and information matrices

The multivariate normal density contains $\det(\Sigma)^{-1/2}$ , so maximum-likelihood estimation works with $\log\det(\Sigma)$ . Multiplicativity turns the determinant of a factorised covariance $\Sigma = LL^{t}$ into $\det(L)^{2}$ , which is the diagonal product of a triangular factor — the practical route to a log-determinant. A singular covariance signals a linearly dependent measurement channel.

Computer graphics and CAD

Volume scaling and orientation of transformations

The determinant of a $3 \times 3$ transformation matrix is the factor by which volumes are scaled, with a negative sign indicating a reflection. Multiplicativity means the volume factor of a chain of transformations is the product of the individual factors, so a composed modelling transform can be checked for degeneracy or handedness reversal without composing the matrices.

Electrical power systems

Load-flow Jacobian degeneracy

The Newton-Raphson load flow iterates on a Jacobian matrix whose determinant approaches zero as the network nears its maximum loadability. Monitoring the determinant — or, in practice, the smallest singular value — provides a voltage-collapse proximity indicator, and the null vector at the singular point identifies the buses that participate in the collapse.

Design Considerations

Use multiplicativity to avoid forming products

When a matrix is defined as a product, a similarity transform or a power, its determinant is available from the factors without ever assembling the product. This is not merely a saving in arithmetic: forming $A B$ explicitly can lose accuracy and destroy sparsity, and multiplicativity avoids both.

Do not use a floating-point determinant as a singularity test

Rounding turns an exact zero determinant into a small non-zero number, and the scaling identity $\det(\alpha A) = \alpha^{n}\det(A)$ makes any absolute threshold meaningless. A well-conditioned $30 \times 30$ matrix of entries near $0.1$ has a determinant near $10^{-30}$ without being remotely singular. Use rank, the condition number, or the smallest singular value.

Reserve the determinant test for exact and symbolic work

Where the entries are integers, rationals or symbols, $\det(A) = 0$ is a clean and decisive test, and the polynomial in a design parameter can be factored to expose every critical value at once. This is where the determinant is unambiguously the right instrument, and it is the setting in which most engineering uses of it actually occur.

Never assume additivity, and be careful with the conjugate transpose

$\det(A+B)$ has no simple form. Separately, note that transpose invariance is exact but the adjoint version carries a conjugation: $\det(A^{*}) = \overline{\det(A)}$ . For a real matrix the two coincide and the distinction is invisible, which is precisely why it surfaces late and unhelpfully in complex-valued work.

Prefer the log-determinant when magnitude is extreme

For large matrices the determinant routinely overflows or underflows the floating-point range even when the matrix is perfectly ordinary. Statistical and machine-learning code should compute the sign and $\log|\det(A)|$ separately from a triangular factor, which is numerically safe and is what the standard library routines provide.

Watch for the difference between a zero determinant and a near-zero one

A determinant that is exactly zero is a structural statement about the model. A determinant that is merely small may be an artefact of scaling, of units, or of rounding. When a determinant comes out small in an engineering model, first re-express the problem in non-dimensional form; a great many apparent near-singularities are units problems.

Standards & Codes

Notation, interchange and numerical standards that govern how this material is written down, stored and computed in production systems.

Applicable standards, conventions and reference implementations
Reference	Title	Relevance to this topic
`ISO 80000-2`	Quantities and units — Part 2: Mathematics	Specifies the upright $\det$ operator, the vertical-bar determinant notation, and the transpose and inverse superscripts used in every identity on this page.
`LAPACK xGETRF / xPOTRF`	LU and Cholesky factorisation routines	The practical route to determinants. Multiplicativity applied to $PA = LU$ or $\Sigma = LL^{t}$ turns the determinant into a diagonal product of the triangular factor times a permutation sign.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Defines the overflow, underflow and rounding behaviour that makes a computed determinant unsuitable as a singularity test and motivates the separate sign-and-logarithm representation.
`IEC 61131-3`	Programmable controllers — Programming languages	Relevant where a small parametric determinant is evaluated inside control logic, for instance a manipulator singularity guard; the standard's fixed-point and real data types govern how the polynomial condition may be implemented on the target.
`ISO 2394`	General principles on reliability for structures	Underpins the structural stability limit states in which a vanishing stiffness determinant defines the critical condition, and frames how the resulting critical loads are treated in a partial-factor design format.

Material Selection

For a mathematical topic, "material" is the numeric representation: the scalar field, storage format and precision the computation is built from.

Representation and precision selection
Representation	Select when	Trade-off
Exact integer arithmetic	Integer matrices where the singularity question must be answered definitively, as in lattice, coding and modular problems.	Gives an unambiguous zero test with no rounding, but the determinant magnitude grows quickly and a fixed-width type will overflow silently.
Exact rational or polynomial arithmetic	Parametric matrices where the determinant is required as a factorable expression in a design variable.	Exposes every critical parameter value at once, but expression swell during expansion can dominate the runtime for larger matrices.
IEEE 754 binary64 via LU factorisation	Numerical determinants of moderate size where the value itself, not a singularity decision, is required.	Cubic cost and backward stability, but the result may overflow for large sizes and must never be used as a rank or conditioning proxy.
Signed logarithm of the determinant	Gaussian likelihoods, evidence computations and any setting where only $\log\|\det\|$ enters the formula.	Immune to overflow and underflow across a very wide range, but loses the magnitude itself and needs a separate branch for an exactly zero determinant.
Cholesky factorisation for symmetric positive definite input	Covariance and stiffness matrices known in advance to be symmetric positive definite.	Half the cost of LU and the factorisation itself fails cleanly if the matrix is not positive definite, but it is inapplicable to general matrices.
Singular value decomposition	The real question is conditioning or numerical rank rather than the determinant value.	Gives the definitive answer, since the determinant magnitude is the product of the singular values and the smallest singular value measures distance to singularity, but costs several times an LU factorisation.

Manufacturing Notes

Implementation notes — how the result is actually produced by hand, by algorithm and by library, including cost and numerical behaviour.

Computing a determinant in practice

Factor $PA = LU$ by Gaussian elimination with partial pivoting, multiply the diagonal entries of $U$ , and multiply by $+1$ or $-1$ according to the parity of the row interchanges recorded in $P$ . This is multiplicativity in action: $\det(P)\det(A) = \det(L)\det(U)$ , with $\det(L) = 1$ because $L$ is unit lower triangular. The cost is about $\tfrac{2}{3}n^3$ operations.

Verifying the multiplicative property by hand

Choose two small matrices with easily computed determinants, form the product, and expand. The check is worth doing once because it is genuinely counter-intuitive: nothing in the entrywise definition of matrix multiplication suggests that a single scalar summary should survive it. Repeat with the factors reversed to see that $\det(BA)$ agrees, even though $BA \ne AB$ .

Library behaviour

numpy.linalg.det returns a float via LU; numpy.linalg.slogdet returns the sign and

\log|\det|

and is the correct choice whenever overflow is plausible. SciPy exposes the same through scipy.linalg.det. SymPy's Matrix.det() works symbolically and is the tool for parametric determinants. MATLAB's det documentation explicitly warns against using the result to test for singularity and directs users to cond or rank.

Symbolic expansion strategy for parametric matrices

Expand about the row or column containing the fewest occurrences of the parameter, not merely the fewest zeros. Keeping the parameter out of as many subdeterminants as possible limits the degree of the intermediate expressions and makes the final factorisation tractable. For matrices of the form $A - \lambda I$ the parameter appears on the diagonal, so a sparse off-diagonal structure is what makes the expansion manageable.

Checking a factorisation-based determinant

Two cheap checks are available. Compute the determinant of the transpose, which must agree exactly in exact arithmetic and to rounding in floating point. Second, if the matrix is a known product, compare the computed determinant against the product of the factors' determinants; disagreement beyond rounding indicates an error in the factorisation rather than in the determinant.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Assuming the determinant is additive	high	Writing $\det(A+B) = \det(A)+\det(B)$ by analogy with linear operations, or with the trace, which really is additive.	Remember that the determinant is multilinear in the rows but not linear in the matrix. Test any proposed identity on $A = I_2$ , $B = -I_2$ .
Using determinant magnitude as a conditioning measure	high	Interpreting a small determinant as near-singularity, ignoring that $\det(\alpha A) = \alpha^{n}\det(A)$ .	Use the condition number or the smallest singular value. If a determinant must be reported, non-dimensionalise the matrix first.
Testing singularity in floating point	high	Comparing a rounded determinant against zero or against a fixed tolerance, which produces both false positives and false negatives.	Use a rank-revealing factorisation, or restrict the determinant test to exact arithmetic where zero means zero.
Overflow or underflow of the determinant	medium	Multiplying $n$ diagonal entries of a triangular factor whose magnitudes are far from one, leaving the representable range.	Use a signed log-determinant routine such as `slogdet` instead of the raw value.
Confusing transpose with conjugate transpose	medium	Applying $\det(A^{t}) = \det(A)$ to the adjoint of a complex matrix, where the correct statement is $\det(A^{*}) = \overline{\det(A)}$ .	Test complex cases explicitly; the two statements are indistinguishable on real data and the error survives real-valued test suites.
Applying determinant identities to non-square or mismatched matrices	medium	Invoking $\det(AB) = \det(A)\det(B)$ when $A$ is $m \times n$ and $B$ is $n \times m$ with $m \ne n$ , where the individual determinants do not exist.	Confirm both factors are square and of the same size before applying multiplicativity.
Solving a system by Cramer's rule at scale	medium	Treating the determinant identities as a practical solution method, requiring $n+1$ determinants where one factorisation would do.	Reserve Cramer's rule for $n \le 3$ or for symbolic derivations; use LU factorisation for numerical solutions.
Missing a repeated root of a parametric determinant	low	Solving $\det(A(k)) = 0$ numerically and reporting distinct roots only, so a repeated critical value is counted once.	Factor the polynomial symbolically and record multiplicities, which carry structural meaning such as coincident buckling modes.

FAQs

Why does the determinant of a product equal the product of the determinants?

The standard argument factors one matrix into elementary matrices, each of which multiplies a determinant by a known scalar — $1$ , $-1$ or $\alpha$ — matching exactly the effect of the corresponding row operation. Applying them one at a time to the second matrix accumulates precisely $\det(A)$ as the total factor. Geometrically, the determinant is a volume scaling factor and composing two maps multiplies their scaling factors.

Does

\det(AB) = \det(BA)

even though

AB \ne BA

Yes, for square matrices of the same size. Both sides equal $\det(A)\det(B)$ , and scalar multiplication is commutative even though matrix multiplication is not. The determinant simply cannot see the ordering, which is one measure of how much information it discards.

Why is a small determinant not evidence of near-singularity?

Because scaling the matrix by $\alpha$ scales the determinant by $\alpha^{n}$ . A $30 \times 30$ matrix whose entries are all around $0.1$ has a determinant of order $10^{-30}$ regardless of how well conditioned it is. Distance to singularity is measured by the smallest singular value, which is scale-aware in the way the determinant is not.

Is the determinant unchanged by a change of basis?

Yes. A change of basis replaces $A$ by $P^{-1}AP$ , and multiplicativity gives $\det(P)^{-1}\det(A)\det(P) = \det(A)$ . The determinant therefore belongs to the underlying linear map rather than to any particular matrix representation of it, which is why it can be used to define the determinant of an operator.

What is the relationship between the determinant and the eigenvalues?

The determinant equals the product of the eigenvalues, counted with algebraic multiplicity. That is consistent with the singularity criterion: a zero determinant means at least one eigenvalue is zero, which means a non-trivial null vector exists. It also explains why the characteristic polynomial's constant term is $\pm\det(A)$ .

Why does transposition leave the determinant alone?

Fully unwound, the determinant is a sum of $n!$ signed products, each taking one entry from every row and every column. Transposing permutes which index is called the row and which the column, mapping the set of products onto itself with matching signs. The inductive proof formalises this by pairing an expansion about row one of $A$ with an expansion about column one of $A^{t}$ .

Can I use

\det(A) \ne 0

to prove a matrix is invertible?

Yes — the singularity criterion together with the theorem that non-singular matrices are invertible makes a non-zero determinant a complete certificate of invertibility, and of full rank, trivial null space, independent columns and unique solutions. In exact arithmetic this is a legitimate proof. In floating point it is not, because the computed determinant is not the true one.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section DM, Subsection PD. Licensed under the GNU Free Documentation License v1.2.
ISO 80000-2:2019, Quantities and units — Part 2: Mathematics. International Organization for Standardization.
Horn, R. A. and Johnson, C. R. Matrix Analysis, 2nd edition. Cambridge University Press, 2013.
Higham, N. J. Accuracy and Stability of Numerical Algorithms, 2nd edition. Society for Industrial and Applied Mathematics, 2002.
Anderson, E. et al. LAPACK Users' Guide, 3rd edition. Society for Industrial and Applied Mathematics, 1999.

AI Suggested Questions

Prove $\det(AB) = \det(A)\det(B)$ by factoring $A$ into elementary matrices, and identify where the hypothesis that both matrices are square is used.
Construct a well-conditioned matrix whose determinant is smaller than $10^{-40}$ and a badly conditioned matrix whose determinant is close to $1$ .
Given a $4 \times 4$ stiffness matrix containing a load factor on the diagonal, expand the determinant symbolically and find all critical load factors with their multiplicities.
Explain why the determinant equals the product of the eigenvalues, and reconcile that with the singularity criterion.
How does the block formula $\det\begin{bmatrix} A & B \\ 0 & D \end{bmatrix} = \det(A)\det(D)$ follow from the properties on this page, and when does it fail for a full block matrix?
Compare determinant, rank and smallest singular value as singularity indicators on a matrix perturbed by rounding at the $10^{-12}$ level.

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Compare determinant, rank, condition number and smallest singular value to decide whether a matrix is truly singular.

Parametric Determinant Solver

Expand the determinant of a matrix containing a symbolic parameter and report the roots at which it becomes singular.