Matrix Determinant Calculator
Exact 2x2 and 3x3 determinants with invertibility verdict.
What it does
A determinant distills a square matrix into one number: the area or volume scale factor of its linear map. Nonzero means the matrix inverts and systems solve uniquely; zero means rows depend on each other and no inverse exists.
The tool above takes 2x2 or 3x3 entries. It returns the exact determinant with the size and an invertible verdict beside it.
The default 3x3 of rows (1,2,3), (0,1,4), (5,6,0) expands to 1 x (0-24) - 2 x (0-20) + 3 x (0-5) = -24 + 40 - 15 = 1: invertible. A 2x2 of (4,6),(3,8) gives 32 - 18 = 14.
Expansion follows Khan Academy linear algebra units with the same cofactor rules in LibreTexts math references. Both treat zero as singular regardless of size.
How to use
- Pick 2x2 or 3x3 first.
- Enter the entries row by row.
- Read the determinant first.
- Read invertible for the yes or no verdict.
- Feed nonzero results into inverse work.
Formula
- ad - bc
- Two by two cross difference.
- cofactors
- Signed 2x2 minors along row one.
- zero
- Singular flag: no inverse.
Near zero values within 1e-9 snap to zero to absorb float dust. Exact decimals entered stay exact through the integer friendly arithmetic.
Worked example
Example: rows (1,2,3), (0,1,4), (5,6,0)
3x3 defaults as shown.
- First term: 1 x (1x0 - 4x6) = 1 x -24 = -24.
- Second: -2 x (0x0 - 4x5) = -2 x -20 = 40.
- Third: 3 x (0x6 - 1x5) = 3 x -5 = -15.
- Total: -24 + 40 - 15 = 1.
- Tool shows 1, 3x3, Yes.
Reference table
| Matrix | det | Verdict |
|---|---|---|
| [4 6; 3 8] | 14 | Invertible |
| [2 4; 1 2] | 0 | Singular |
| Example 3x3 | 1 | Invertible |
| Identity 3x3 | 1 | Invertible |
Limits
- 2x2 and 3x3 only; larger needs software.
- Square matrices only by definition.
- Near zero snaps within 1e-9 tolerance.
- Entries cap at 1e9 magnitude.
Common mistakes
- Sign errors on the middle cofactor.
- Reading zero as invertible.
- Mixing row and column order.
- Rounding minors before combining.
- Applying determinants to non square matrices.
Try it yourself
Quick checks. Answers sit tucked under each question.
2x2 [4 6; 3 8]?
4x8 - 6x3 = 32 - 18 = 14.
2x2 [2 4; 1 2]?
4 - 4 = 0: singular, no inverse.
Identity det?
Always 1 at any size.
Questions
What is a determinant?
One number scaling areas (2x2) or volumes (3x3) under the matrix map.
How do you compute 3x3?
Cofactor expansion: a(ei-fh) - b(di-fg) + c(dh-eg).
What does zero mean?
Singular: rows depend, systems lack unique answers, no inverse.
Why alternate signs?
Checkerboard cofactor pattern (+ - +) along the row.
Determinant of identity?
1: it scales nothing.
Bigger matrices?
Row reduction or software; cofactor cost grows factorially.