\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bcVariables
- a, b, c, d: matrix entries
- det: scalar determinant
How to use this formula
Calculates the determinant of a two-by-two matrix.
Important notes
- A zero determinant means the matrix is singular and has no inverse.
- Keep the subtraction order ad − bc.
Quick example
The determinant of [[1,2],[3,4]] is −2.
Applicability, worked calculation, and verification
Domain and applicability
Applies to a 2 × 2 square matrix; a zero determinant indicates singularity and no two-sided inverse.
Units
- determinant units are the product of one entry from each row and column
Assumptions and domain checks
- A zero determinant means the matrix is singular and has no inverse.
- Preserve matrix order and verify dimension compatibility; multiplication and inversion are not generally commutative or always defined.
- Matrix and vector dimensions must be compatible, and any required inverse, determinant, rank, or basis condition must hold.
Do not use this formula when
- Do not use 2×2 Matrix Determinant when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For 2×2 Matrix Determinant, check zero, negative, and extreme input values before relying on the result.
- When using 2×2 Matrix Determinant, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form \det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For the matrix [[3, 2], [1, 4]], det(A) = 3·4 − 2·1 = 12 − 2 = 10.
- Identify a = 3, b = 2, c = 1, and d = 4.
- Multiply the main diagonal entries: ad = 3 × 4 = 12.
- Multiply the other diagonal entries: bc = 2 × 1 = 2.
- Subtract to obtain det(A) = 12 − 2 = 10.
Independent verification
Because the determinant is nonzero, the matrix is invertible; direct multiplication by its computed inverse returns the identity matrix.
Common mistakes
- Do not reorder factors or mix incompatible dimensions when using 2×2 Matrix Determinant; matrix operations are order-sensitive.
- Do not assume commutativity, invertibility, independence, or full rank unless the required condition has been established.
Continue the workflow
Use 2×2 Matrix Determinant in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for 2×2 Matrix Determinant.
Verified against: OpenStax Algebra and Trigonometry; NIST Digital Library of Mathematical Functions
Automated quality check: Passed the core formula indexing gate.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the 2×2 Matrix Determinant used for?
Calculates the determinant of a two-by-two matrix.
Can I copy this formula as LaTeX?
Yes. Copy \det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.