Determinants and Invertibility: How Linear Algebra Reveals Reversible Structure

Last Updated June 28, 2026

Determinants and invertibility explain when a square matrix preserves enough structure for a transformation to be reversed. The determinant is not just a number produced by a formula. It summarizes volume scaling, orientation, singularity, dependency, and whether a linear transformation collapses space into a lower-dimensional structure.

This article introduces determinants as structural diagnostics for square matrices. It connects determinant value, signed volume, orientation, invertibility, singular matrices, full rank, pivot structure, row dependence, column dependence, transformation geometry, numerical caution, and systems modeling interpretation.

The central modeling question is not only “What is the determinant?” It is “Does this transformation preserve enough independent structure to support recovery, reversibility, or stable interpretation?”

Determinants sit at the intersection of algebra, geometry, and systems interpretation. Algebraically, the determinant helps determine whether a square matrix is invertible. Geometrically, it describes how a linear transformation scales signed volume. Structurally, it indicates whether independent directions remain independent or collapse into dependence.

For systems modeling, determinants are useful because they reveal whether a transformation preserves enough structure to recover inputs from outputs. A nonzero determinant means the matrix is invertible and full rank. A zero determinant means the matrix is singular: some dimension of structure has collapsed, and exact recovery is not possible through that matrix alone.

Why Determinants Matter

Determinants matter because they summarize whether a square matrix preserves independent structure. A determinant can tell whether a transformation expands or contracts volume, preserves or reverses orientation, and remains reversible or collapses dimensions.

In linear systems, a nonzero determinant means a square coefficient matrix has full rank and an inverse. A zero determinant means the matrix is singular. Some independent direction has been lost, and the system cannot uniquely recover inputs from outputs.

For systems modeling, this makes the determinant a structural warning signal. It helps identify whether a model transformation is reversible, whether variables are independently identifiable, whether a coefficient system has a unique solution, and whether a model has collapsed distinct system states into the same output.

The determinant is therefore not only a formula. It is a compact structural diagnostic.

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What Is a Determinant?

The determinant is a scalar associated with a square matrix. It is defined only for square matrices because it describes how a linear transformation maps an \(n\)-dimensional space to itself.

For a \(2\times2\) matrix, the determinant has a simple formula:

For larger matrices, determinants can be computed through expansion, row reduction, LU decomposition, or numerical routines. But the interpretation remains the same: determinants measure how the transformation changes signed volume and whether the transformation collapses independent directions.

The determinant compresses a great deal of structural information into one scalar, but that scalar must be interpreted carefully.

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Determinants as Volume Scaling

Geometrically, the determinant tells how much a linear transformation scales volume. In two dimensions, it scales area. In three dimensions, it scales volume. In higher dimensions, it scales \(n\)-dimensional volume.

If \(|\det(A)|=1\), the transformation preserves volume. If \(|\det(A)|>1\), it expands volume. If \(0<|\det(A)|<1\), it contracts volume. If \(\det(A)=0\), it collapses volume to zero.

This volume interpretation helps explain why determinants are linked to invertibility. If volume collapses to zero, information has been lost and exact reversal is impossible.

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Orientation and Sign

The sign of the determinant indicates whether the transformation preserves or reverses orientation. A positive determinant preserves orientation. A negative determinant reverses orientation. A zero determinant collapses orientation because the transformed volume is zero.

In two dimensions, a negative determinant can be understood as a flip of the plane. In three dimensions, it reverses handedness. In systems modeling, orientation is often less directly interpretable than invertibility, but it can still matter when coordinate systems, transformations, projections, or physical geometry are part of the model.

The sign is part of the determinant’s geometric meaning. The magnitude tells how much volume changes; the sign tells whether orientation is preserved.

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Zero Determinants and Collapse

A determinant of zero means the transformation collapses space into a lower-dimensional structure. In two dimensions, a square region might collapse into a line. In three dimensions, a volume might collapse into a plane or line. In higher dimensions, independent directions become dependent.

For systems modeling, this collapse can mean that different inputs produce the same output. Some variable direction has become invisible to the transformation. As a result, exact recovery of the original state is not possible from the transformed output alone.

Zero determinants are not just computational results. They indicate structural loss.

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Invertibility

A square matrix is invertible if there exists another matrix \(A^{-1}\) such that multiplying by \(A^{-1}\) reverses the effect of \(A\). Invertibility means the transformation is one-to-one and onto in the same finite-dimensional space.

For a square matrix, invertibility is equivalent to a nonzero determinant, full rank, a pivot in every row and column, a trivial null space, and unique solutions for every right-hand side.

Invertibility is a strong structural property. It says the transformation preserves enough information to be reversed algebraically.

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Rank, Pivots, and Determinants

Determinants are closely connected to rank and pivot structure. A square matrix has a nonzero determinant exactly when it has full rank. Through row reduction, this means a pivot appears in every row and every column.

This connection is especially useful in systems modeling. Rank and pivots reveal where structure is independent. The determinant summarizes whether full-dimensional independence survives. If rank is deficient, the determinant is zero.

The determinant is therefore one face of a larger structural picture. Rank, pivots, nullity, and determinant all tell related stories about independence and collapse.

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Row Operations and Determinants

Row operations affect determinants in predictable ways. Swapping two rows changes the sign of the determinant. Multiplying a row by a scalar multiplies the determinant by that scalar. Adding a multiple of one row to another row does not change the determinant.

This is why row reduction can be used to compute determinants. If a matrix is reduced to triangular form, the determinant is related to the product of the diagonal entries, adjusted for row swaps and row scalings.

Row-operation tracking helps connect determinant computation to Gaussian elimination and pivot structure.

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Determinants and Linear Systems

For a square linear system \(A\mathbf{x}=\mathbf{b}\), the determinant of \(A\) tells whether the system has a unique solution for every possible right-hand side. If \(\det(A)\neq0\), the system has exactly one solution for every \(\mathbf{b}\). If \(\det(A)=0\), the system may have no solution or infinitely many solutions depending on \(\mathbf{b}\).

This connection matters in systems modeling because many applied systems are written as square coefficient systems. The determinant helps reveal whether the system is structurally recoverable before computing a solution.

The determinant is most useful here as a structural test, not as the preferred computational method for solving large systems.

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Determinants and Structural Dependence

A zero determinant indicates structural dependence in a square matrix. Rows or columns are linearly dependent. At least one row or column fails to add independent information. The transformation maps distinct inputs to overlapping outputs.

In systems modeling, structural dependence may reflect redundancy, repeated indicators, conservation laws, omitted variables, or unidentifiable parameters. A determinant of zero does not automatically mean the model is useless. It means the model does not support full-dimensional recovery through that matrix.

The determinant can identify dependence, but interpretation requires knowing what rows and columns mean.

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Determinants in Higher Dimensions

In higher dimensions, determinants continue to measure signed volume scaling. The intuition becomes harder to visualize, but the structural meaning remains strong. A nonzero determinant means the transformation maps a full-dimensional unit volume into another full-dimensional volume. A zero determinant means the transformed volume collapses into a lower-dimensional structure.

This higher-dimensional view is important in systems modeling because many models involve dozens, hundreds, or thousands of variables. In those cases, determinants may be difficult to interpret directly as a magnitude, but the zero-versus-nonzero distinction remains structurally important.

In high-dimensional systems, determinants often work best as part of a larger diagnostic toolkit that also includes rank, conditioning, singular values, and residual checks.

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Numerical Determinants and Conditioning

Determinants can be numerically delicate. A determinant may be very large or very small because of dimension, scaling, or units. A near-zero determinant may indicate near singularity, but determinant magnitude alone can be misleading without conditioning and scale review.

In computational workflows, determinants are often computed through LU factorization or related methods rather than direct formula expansion. For large matrices, log determinants may be used to avoid overflow or underflow.

Numerical determinant analysis should be paired with rank, conditioning, scaling, and model interpretation. In applied work, “nonzero” is not always enough.

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Systems Modeling Interpretation

Determinants help systems modelers understand whether a square transformation preserves structure. They reveal whether a system transformation is reversible, whether independent directions remain independent, whether variables can be recovered from outputs, and whether hidden collapse or dependence is present.

In infrastructure modeling, a zero determinant may indicate that network equations fail to determine all flows. In economic systems, singularity may reveal sector dependencies or repeated structural relationships. In ecological modeling, a near-zero determinant may indicate fragile equilibrium structure. In policy models, determinant diagnostics may reveal whether indicators or constraints collapse into redundant dimensions.

Responsible interpretation should connect determinant results to row meanings, column meanings, units, scaling, rank, nullity, conditioning, and practical feasibility. A determinant is mathematically precise only within the matrix supplied.

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Mathematical Deepening

This section adds a more formal layer for mathematically advanced readers. Determinants connect multilinearity, alternating forms, bases, orientation, volume forms, eigenvalues, characteristic polynomials, invertibility, LU factorization, singularity, and numerical conditioning.

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Computation and Reproducible Workflows

Computational workflows for determinants and invertibility should document the original matrix, row and column meanings, determinant value, determinant sign, determinant magnitude, rank, nullity, invertibility status, condition estimate, tolerance rules, and interpretation warnings. A determinant should not be reported without context.

The companion repository treats determinant analysis as auditable mathematical modeling. Python, R, Julia, SQL, Haskell, C, C++, Fortran, Rust, Go, notebooks, schemas, generated outputs, Canvas artifacts, advanced reports, and calculators each support a different layer of reproducible systems modeling.

For this article, the computational examples focus on determinant value, invertibility classification, rank comparison, condition warnings, and model-governance records.

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Python Workflow: Determinant and Invertibility Audit

The Python workflow below computes determinant, rank, nullity, invertibility status, condition number, and interpretation warning for a square coefficient matrix.

from __future__ import annotations

from dataclasses import asdict, dataclass
from pathlib import Path
import csv
import json
import math


Matrix = list[list[float]]


@dataclass(frozen=True)
class DeterminantAudit:
    system_name: str
    matrix_size: int
    determinant: float
    determinant_sign: str
    invertible: bool
    rank: int
    nullity: int
    condition_warning: str
    tolerance: float
    interpretation_warning: str


def determinant_3x3(A: Matrix) -> float:
    return (
        A[0][0] * (A[1][1] * A[2][2] - A[1][2] * A[2][1])
        - A[0][1] * (A[1][0] * A[2][2] - A[1][2] * A[2][0])
        + A[0][2] * (A[1][0] * A[2][1] - A[1][1] * A[2][0])
    )


def rref(matrix: Matrix, tolerance: float = 1e-10) -> tuple[Matrix, list[int]]:
    rows = [row[:] for row in matrix]
    row_count = len(rows)
    column_count = len(rows[0]) if rows else 0
    pivot_columns: list[int] = []
    pivot_row = 0

    for column in range(column_count):
        pivot = None
        for row in range(pivot_row, row_count):
            if abs(rows[row][column]) > tolerance:
                pivot = row
                break

        if pivot is None:
            continue

        rows[pivot_row], rows[pivot] = rows[pivot], rows[pivot_row]
        pivot_value = rows[pivot_row][column]
        rows[pivot_row] = [value / pivot_value for value in rows[pivot_row]]

        for row in range(row_count):
            if row != pivot_row:
                factor = rows[row][column]
                rows[row] = [
                    current - factor * pivot_current
                    for current, pivot_current in zip(rows[row], rows[pivot_row])
                ]

        pivot_columns.append(column)
        pivot_row += 1

        if pivot_row == row_count:
            break

    return rows, pivot_columns


def determinant_sign(value: float, tolerance: float) -> str:
    if abs(value) <= tolerance:
        return "zero"
    if value > 0:
        return "positive"
    return "negative"


def build_audit() -> DeterminantAudit:
    A = [
        [1.0, 1.0, 0.0],
        [0.0, 1.0, 1.0],
        [1.0, 0.0, 1.0],
    ]

    tolerance = 1e-10
    det_value = determinant_3x3(A)
    _, pivots = rref(A, tolerance=tolerance)
    matrix_size = len(A)
    rank = len(pivots)
    nullity = matrix_size - rank
    invertible = abs(det_value) > tolerance and rank == matrix_size

    if abs(det_value) <= tolerance:
        condition_warning = "singular or effectively singular under tolerance"
    elif abs(det_value) < 1e-6:
        condition_warning = "near-zero determinant; check conditioning and singular values"
    else:
        condition_warning = "determinant is nonzero; still review conditioning for applied use"

    return DeterminantAudit(
        system_name="three_constraint_recovery_matrix",
        matrix_size=matrix_size,
        determinant=round(det_value, 10),
        determinant_sign=determinant_sign(det_value, tolerance),
        invertible=invertible,
        rank=rank,
        nullity=nullity,
        condition_warning=condition_warning,
        tolerance=tolerance,
        interpretation_warning=(
            "A nonzero determinant indicates algebraic invertibility, but practical recovery "
            "requires review of units, conditioning, data quality, and model purpose."
        ),
    )


def write_outputs(output_dir: Path) -> None:
    (output_dir / "tables").mkdir(parents=True, exist_ok=True)
    (output_dir / "json").mkdir(parents=True, exist_ok=True)

    audit = build_audit()
    row = asdict(audit)

    with (output_dir / "tables" / "determinant_invertibility_audit.csv").open(
        "w", newline="", encoding="utf-8"
    ) as handle:
        writer = csv.DictWriter(handle, fieldnames=list(row.keys()))
        writer.writeheader()
        writer.writerow(row)

    (output_dir / "json" / "determinant_invertibility_audit.json").write_text(
        json.dumps(row, indent=2, sort_keys=True),
        encoding="utf-8",
    )


if __name__ == "__main__":
    write_outputs(Path("outputs"))
    print("Determinant and invertibility audit complete.")

This workflow treats the determinant as an auditable model diagnostic. It records determinant value, sign, rank, nullity, invertibility, tolerance, and interpretation warnings.

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R Workflow: Determinant and Rank Diagnostics

R can support determinant and invertibility diagnostics using determinant, rank, and conditioning checks. The workflow below records determinant, rank, nullity, and interpretive warnings.

A <- matrix(
  c(
    1, 1, 0,
    0, 1, 1,
    1, 0, 1
  ),
  nrow = 3,
  byrow = TRUE
)

tolerance <- 1e-10
det_value <- det(A)
rank_A <- qr(A)$rank
matrix_size <- nrow(A)
nullity_A <- ncol(A) - rank_A

determinant_sign <- if (abs(det_value) <= tolerance) {
  "zero"
} else if (det_value > 0) {
  "positive"
} else {
  "negative"
}

invertible <- abs(det_value) > tolerance && rank_A == matrix_size

condition_warning <- if (abs(det_value) <= tolerance) {
  "singular or effectively singular under tolerance"
} else if (abs(det_value) < 1e-6) {
  "near-zero determinant; check conditioning and singular values"
} else {
  "nonzero determinant; still review conditioning for applied use"
}

audit_record <- data.frame(
  system_name = "three_constraint_recovery_matrix",
  matrix_size = matrix_size,
  determinant = det_value,
  determinant_sign = determinant_sign,
  invertible = invertible,
  rank = rank_A,
  nullity = nullity_A,
  condition_warning = condition_warning,
  tolerance = tolerance,
  interpretation_warning = paste(
    "A determinant is a structural diagnostic;",
    "practical recovery depends on units, conditioning, and model purpose."
  )
)

dir.create("outputs/tables", recursive = TRUE, showWarnings = FALSE)

write.csv(
  audit_record,
  "outputs/tables/r_determinant_invertibility_audit.csv",
  row.names = FALSE
)

print(audit_record)

This R workflow is useful when determinant diagnostics support statistical modeling, matrix computation, calibration, simulation, or policy analysis.

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Haskell Workflow: Typed Determinant Records

Haskell can represent determinant and invertibility audits with explicit fields for determinant value, sign, rank, nullity, invertibility, tolerance, and interpretation warning.

module Main where

data DeterminantAudit = DeterminantAudit
  { systemName :: String
  , matrixSize :: Int
  , determinantValue :: Double
  , determinantSign :: String
  , invertible :: Bool
  , rankValue :: Int
  , nullityValue :: Int
  , tolerance :: Double
  , interpretationWarning :: String
  } deriving (Show)

buildAudit :: DeterminantAudit
buildAudit =
  DeterminantAudit
    "three_constraint_recovery_matrix"
    3
    2.0
    "positive"
    True
    3
    0
    1.0e-10
    "A nonzero determinant indicates algebraic invertibility, but practical recovery requires model review."

main :: IO ()
main =
  print buildAudit

The typed workflow keeps determinant conclusions attached to modeling context rather than treating them as isolated scalar outputs.

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SQL Workflow: Invertibility Assumption Registry

SQL can document determinant and invertibility assumptions when square matrix diagnostics support dashboards, planning systems, scientific-computing pipelines, optimization workflows, risk models, or institutional reports.

CREATE TABLE determinant_invertibility_assumption_registry (
    assumption_key TEXT PRIMARY KEY,
    assumption_name TEXT NOT NULL,
    mathematical_role TEXT NOT NULL,
    systems_modeling_role TEXT NOT NULL,
    review_warning TEXT NOT NULL
);

INSERT INTO determinant_invertibility_assumption_registry VALUES
(
  'determinant',
  'Determinant',
  'Measures signed volume scaling for a square matrix.',
  'Summarizes whether a transformation expands, contracts, reverses, or collapses state-space volume.',
  'Determinant magnitude depends on units, scaling, and dimension.'
);

INSERT INTO determinant_invertibility_assumption_registry VALUES
(
  'nonzero_determinant',
  'Nonzero determinant',
  'Indicates that a square matrix is invertible.',
  'Suggests algebraic recovery of inputs from outputs is possible.',
  'Algebraic recovery does not guarantee practical recovery or stable computation.'
);

INSERT INTO determinant_invertibility_assumption_registry VALUES
(
  'zero_determinant',
  'Zero determinant',
  'Indicates singularity and rank deficiency.',
  'Signals structural collapse, dependence, or non-identifiability.',
  'Singularity should be interpreted using row meanings, column meanings, and model purpose.'
);

INSERT INTO determinant_invertibility_assumption_registry VALUES
(
  'orientation',
  'Orientation',
  'The sign of the determinant indicates orientation preservation or reversal.',
  'Shows whether the transformation preserves or flips geometric orientation.',
  'Orientation may or may not have direct system meaning depending on the model.'
);

INSERT INTO determinant_invertibility_assumption_registry VALUES
(
  'near_singularity',
  'Near singularity',
  'A near-zero determinant or small singular values indicate numerical fragility.',
  'Warns that small data changes may produce large solution changes.',
  'Check conditioning, scaling, tolerance, and singular values.'
);

INSERT INTO determinant_invertibility_assumption_registry VALUES
(
  'invertibility',
  'Invertibility',
  'Equivalent to nonzero determinant, full rank, and trivial null space for square matrices.',
  'Indicates full structural recoverability under the matrix model.',
  'Invertibility is formal; real-world reversibility requires additional justification.'
);

SELECT
    assumption_name,
    mathematical_role,
    systems_modeling_role,
    review_warning
FROM determinant_invertibility_assumption_registry
ORDER BY assumption_key;

This registry keeps determinant analysis tied to volume scaling, orientation, rank, nullity, conditioning, and responsible interpretation.

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GitHub Repository

The companion repository for this article is designed as a reproducible mathematical-modeling workspace. It supports determinant audits, invertibility diagnostics, rank comparisons, nullity checks, singularity warnings, condition-review notes, SQL governance tables, generated outputs, advanced mathematical audit reports, and reusable calculator scripts.

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Interpretive Limits and Responsible Use

Determinants are powerful because they summarize signed volume scaling, orientation, singularity, and invertibility. They are limited because they apply only to square matrices and compress many structural features into one scalar. A determinant alone cannot validate a model, justify assumptions, guarantee feasibility, or ensure numerical stability.

A nonzero determinant means algebraic invertibility, not necessarily practical reversibility. A zero determinant means structural collapse, but that collapse may be meaningful, expected, or useful. A near-zero determinant may warn of numerical fragility, but conditioning and singular values are needed for responsible computational interpretation. Determinant magnitude depends on units and scaling, so it should not be interpreted without context.

Responsible use requires documenting matrix meaning, row and column definitions, units, determinant method, rank, nullity, condition estimates, tolerance choices, and the difference between formal invertibility and real-world recoverability.

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Related Articles

• What Is Linear Algebra for Systems Modeling?
• Scalars, Vectors, and System States
• Vector Spaces and System Representation
• Span, Linear Independence, and Basis
• Dimension and the Structure of Solution Spaces
• Matrices and the Organization of Multivariable Systems
• Matrix Arithmetic and the Logic of Combination
• Systems of Linear Equations
• Gaussian Elimination and Row Reduction
• Pivot Structure and Solvability
• Rank, Nullity, and Structural Dependence
• Linear Algebra for Systems Modeling
• Mathematical Modeling
• Systems Modeling
• Scientific Computing for Systems Modeling

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Further Reading

• Axler, S. (2024) Linear Algebra Done Right. 4th edn. Cham: Springer. Available at: https://linear.axler.net/.
• Boyd, S. and Vandenberghe, L. (2018) Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares. Cambridge: Cambridge University Press. Available at: https://vmls-book.stanford.edu/.
• Boyd, S. and Vandenberghe, L. (2018) Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares. PDF edition. Stanford University and UCLA. Available at: https://web.stanford.edu/~boyd/vmls/vmls.pdf.
• Georgia Institute of Technology (n.d.) Interactive Linear Algebra. Georgia Institute of Technology. Available at: https://textbooks.math.gatech.edu/ila/.
• Hefferon, J. (n.d.) Linear Algebra. Saint Michael’s College. Available at: https://joshua.smcvt.edu/linearalgebra/.
• Massachusetts Institute of Technology OpenCourseWare (2011) Linear Algebra. Cambridge, MA: MIT OpenCourseWare. Available at: https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/.
• Massachusetts Institute of Technology OpenCourseWare (2018) Matrix Methods in Data Analysis, Signal Processing, and Machine Learning. Cambridge, MA: MIT OpenCourseWare. Available at: https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/.
• MIT Mathematics (2023) Introduction to Linear Algebra, Sixth Edition. Cambridge, MA: Massachusetts Institute of Technology. Available at: https://math.mit.edu/~gs/linearalgebra/ila6/indexila6.html.
• Strang, G. (2016) Introduction to Linear Algebra. Wellesley, MA: Wellesley-Cambridge Press. Available at: https://math.mit.edu/~gs/linearalgebra/.
• Treil, S. (2017) Linear Algebra Done Wrong. Providence, RI: Brown University. Available at: https://www.math.brown.edu/streil/papers/LADW/LADW.html.
• van de Geijn, R. and Myers, M. (n.d.) Linear Algebra: Foundations to Frontiers. Austin, TX: The University of Texas at Austin. Available at: https://www.cs.utexas.edu/~flame/laff/laff/LAFF-2.00M.pdf.
• Golub, G.H. and Van Loan, C.F. (2013) Matrix Computations. 4th edn. Baltimore, MD: Johns Hopkins University Press. Available at: https://www.press.jhu.edu/books/title/10678/matrix-computations.
• Higham, N.J. (2002) Accuracy and Stability of Numerical Algorithms. 2nd edn. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://epubs.siam.org/doi/book/10.1137/1.9780898718027.
• Horn, R.A. and Johnson, C.R. (2013) Matrix Analysis. 2nd edn. Cambridge: Cambridge University Press. Available at: https://www.cambridge.org/highereducation/books/matrix-analysis/FDA3627DC2B9F5C3DF2FD8C3CC136B48.
• Meyer, C.D. (2023) Matrix Analysis and Applied Linear Algebra. 2nd edn. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://epubs.siam.org/doi/book/10.1137/1.9781611977448.
• Trefethen, L.N. and Bau, D. (1997) Numerical Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://epubs.siam.org/doi/book/10.1137/1.9780898719574.
• National Institute of Standards and Technology (n.d.) Matrix Market. NIST Mathematical and Computational Sciences Division. Available at: https://math.nist.gov/MatrixMarket/.
• Netlib (n.d.) BLAS — Basic Linear Algebra Subprograms. Netlib Repository. Available at: https://www.netlib.org/blas/.
• Netlib (n.d.) LAPACK — Linear Algebra PACKage. Netlib Repository. Available at: https://www.netlib.org/lapack/.
• Netlib and SIAM (1999) LAPACK Users’ Guide. 3rd edn. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://www.netlib.org/lapack/lug/.
• NumPy Developers (n.d.) Linear Algebra: numpy.linalg. NumPy Documentation. Available at: https://numpy.org/doc/stable/reference/routines.linalg.html.
• SciPy Developers (n.d.) Linear Algebra: scipy.linalg. SciPy Documentation. Available at: https://docs.scipy.org/doc/scipy/reference/linalg.html.
• SciPy Developers (n.d.) LU Factorization: scipy.linalg.lu_factor. SciPy Documentation. Available at: https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.lu_factor.html.
• SciPy Developers (n.d.) Matrix Determinant: scipy.linalg.det. SciPy Documentation. Available at: https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.det.html.
• R Core Team and Matrix Package Authors (n.d.) Matrix: Sparse and Dense Matrix Classes and Methods. CRAN. Available at: https://cran.r-project.org/package=Matrix.
• Julia Documentation (n.d.) Linear Algebra — Julia Standard Library. Julia Documentation. Available at: https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/.

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References

• Axler, S. (2024) Linear Algebra Done Right. 4th edn. Cham: Springer. Available at: https://linear.axler.net/.
• Boyd, S. and Vandenberghe, L. (2018) Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares. Cambridge: Cambridge University Press. Available at: https://vmls-book.stanford.edu/.
• Georgia Institute of Technology (n.d.) Interactive Linear Algebra. Georgia Institute of Technology. Available at: https://textbooks.math.gatech.edu/ila/.
• Golub, G.H. and Van Loan, C.F. (2013) Matrix Computations. 4th edn. Baltimore, MD: Johns Hopkins University Press. Available at: https://www.press.jhu.edu/books/title/10678/matrix-computations.
• Hefferon, J. (n.d.) Linear Algebra. Saint Michael’s College. Available at: https://joshua.smcvt.edu/linearalgebra/.
• Higham, N.J. (2002) Accuracy and Stability of Numerical Algorithms. 2nd edn. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://epubs.siam.org/doi/book/10.1137/1.9780898718027.
• Horn, R.A. and Johnson, C.R. (2013) Matrix Analysis. 2nd edn. Cambridge: Cambridge University Press. Available at: https://www.cambridge.org/highereducation/books/matrix-analysis/FDA3627DC2B9F5C3DF2FD8C3CC136B48.
• Julia Documentation (n.d.) Linear Algebra — Julia Standard Library. Julia Documentation. Available at: https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/.
• Massachusetts Institute of Technology OpenCourseWare (2011) Linear Algebra. Cambridge, MA: MIT OpenCourseWare. Available at: https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/.
• Massachusetts Institute of Technology OpenCourseWare (2018) Matrix Methods in Data Analysis, Signal Processing, and Machine Learning. Cambridge, MA: MIT OpenCourseWare. Available at: https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/.
• Meyer, C.D. (2023) Matrix Analysis and Applied Linear Algebra. 2nd edn. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://epubs.siam.org/doi/book/10.1137/1.9781611977448.
• MIT Mathematics (2023) Introduction to Linear Algebra, Sixth Edition. Cambridge, MA: Massachusetts Institute of Technology. Available at: https://math.mit.edu/~gs/linearalgebra/ila6/indexila6.html.
• Netlib (n.d.) BLAS — Basic Linear Algebra Subprograms. Netlib Repository. Available at: https://www.netlib.org/blas/.
• Netlib (n.d.) LAPACK — Linear Algebra PACKage. Netlib Repository. Available at: https://www.netlib.org/lapack/.
• Netlib and SIAM (1999) LAPACK Users’ Guide. 3rd edn. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://www.netlib.org/lapack/lug/.
• NumPy Developers (n.d.) Linear Algebra: numpy.linalg. NumPy Documentation. Available at: https://numpy.org/doc/stable/reference/routines.linalg.html.
• R Core Team and Matrix Package Authors (n.d.) Matrix: Sparse and Dense Matrix Classes and Methods. CRAN. Available at: https://cran.r-project.org/package=Matrix.
• SciPy Developers (n.d.) Linear Algebra: scipy.linalg. SciPy Documentation. Available at: https://docs.scipy.org/doc/scipy/reference/linalg.html.
• SciPy Developers (n.d.) Matrix Determinant: scipy.linalg.det. SciPy Documentation. Available at: https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.det.html.
• Strang, G. (2016) Introduction to Linear Algebra. Wellesley, MA: Wellesley-Cambridge Press. Available at: https://math.mit.edu/~gs/linearalgebra/.
• Treil, S. (2017) Linear Algebra Done Wrong. Providence, RI: Brown University. Available at: https://www.math.brown.edu/streil/papers/LADW/LADW.html.
• Trefethen, L.N. and Bau, D. (1997) Numerical Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://epubs.siam.org/doi/book/10.1137/1.9780898719574.
• van de Geijn, R. and Myers, M. (n.d.) Linear Algebra: Foundations to Frontiers. Austin, TX: The University of Texas at Austin. Available at: https://www.cs.utexas.edu/~flame/laff/laff/LAFF-2.00M.pdf.

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