LaTeX
\det A=a(ei-fh)-b(di-fg)+c(dh-eg)Variables
- a through i: matrix entries
How to use this formula
Expands the determinant of a 3×3 matrix along the first row.
Important notes
- Cofactor signs alternate +,−,+.
Quick example
Use row operations for larger matrices.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Cofactor signs alternate +,−,+.
- 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.
Worked example
Use row operations for larger matrices.
Common mistakes
- Do not reorder factors or mix incompatible dimensions when using Three by Three 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 Three by Three 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 A=a(ei-fh)-b(di-fg)+c(dh-eg). - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- 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 Three by Three Determinant used for?
Expands the determinant of a 3×3 matrix along the first row.
Can I copy this formula as LaTeX?
Yes. Copy \det A=a(ei-fh)-b(di-fg)+c(dh-eg) 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.