Matrix Determinant Calculator
Calculate the determinant of a 3×3 matrix.
Last updated · View algorithm, constants and default inputs (JSON)
About this calculator
The determinant tells you whether a matrix can be inverted and how it scales volume. If the determinant is zero the matrix is singular, meaning it squashes space into a lower dimension.
Worked example
calculated with the default values below
a 2 · b 1 · c 3 · d 0 · e 4 · f 1 · g 5 · h 2 · i 0 Determinant -59.000000 · Trace 6.0000 · Singular (not invertible) No
General background references
Disclaimer
These calculators give general mathematical results. They suit study, planning and estimation — verify anything that feeds a graded submission, a published figure or a financial commitment.
Implementation source
This record describes the implementation used by this page. It is not independent validation. External references, when available, support only the stated formula or definition; estimates and custom rules retain their own assumptions.
Formula and variables
det = a(ei − fh) − b(di − fg) + c(dh − eg); tr = a + e + i
- det
- determinant of A
- tr
- trace
- A
- 3×3 matrix [[a,b,c],[d,e,f],[g,h,i]]
Cofactor expansion along the first row. The matrix is singular when |det| < 1e-12.
Frequently asked questions
Why does my matrix show singular when the determinant is not exactly zero?
The tool flags singular when the absolute determinant is below 1e-12, because floating-point math rarely lands exactly on zero. The 3 by 3 determinant uses a times e i minus f h, minus b times d i minus f g, plus c times d h minus e g, so a row that is an exact multiple of another gives a tiny but nonzero result that still counts as not invertible. The Trace output is simply a plus e plus i and is unaffected by this threshold.