Determinant of A
-2
Signed determinant of the coefficient matrix.
Mathematics · Vectors & matrices
Matrix calculator — determinant, inverse and operations
Calculate determinants, inverses, sums, differences and products of 2×2 or 3×3 matrices with structured matrix results.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
03
Results
Results show up to four significant digits for readability; internal calculations retain full precision.
See calculation details
01Determinant of A
Formula
det A = a₁₁a₂₂ − a₁₂a₂₁Substitution
1 × 4 − 2 × 3Result
-2
01Formulas and symbols
Formulas used
R
det A₂×₂ = a₁₁a₂₂ − a₁₂a₂₁
AA⁻¹ = I
det A: determinant; A⁻¹: inverse; I: identity matrix.
R
(AB)ᵢⱼ = ∑ₖ aᵢₖbₖⱼ
(A±B)ᵢⱼ = aᵢⱼ±bᵢⱼ
aᵢⱼ and bᵢⱼ: coefficients in row i and column j.
| Symbol | Meaning |
|---|---|
| n | Dimension |
| A | Matrix A |
| B | Matrix B |
| det A | Determinant of A |
| R | Result matrix |
02Validation example
Reference numerical case
- A = [[1, 2], [3, 4]] has determinant −2 and inverse [[−2, 1], [1.5, −0.5]]. Multiplying A by its inverse gives the identity matrix.
03How it works
Enter one value in each cell of the matrix grid. Addition and subtraction use corresponding entries. Multiplication sums row-by-column products. Inversion applies partial-pivot Gauss–Jordan elimination to [A|I].
The entry in row i, column j of AB is the dot product of row i of A with column j of B. Addition and subtraction operate on matching entries.
04Limits and conventions
Square real matrices of order 2 or 3, with compatible dimensions. A singular matrix has no inverse. Numerical pivots below 64 machine epsilons after row scaling are treated as zero; extremely ill-conditioned inverses are not reliable.
05Common mistakes
Matrix multiplication is generally not commutative: AB and BA can differ. The inverse is not the entrywise reciprocal.
06Practical questions
When does a matrix have no inverse?
A square matrix has no inverse when it is singular, which is equivalent to det(A) = 0. Its rows or columns are then linearly dependent.
What does a zero determinant mean?
A zero determinant means the matrix is singular. Geometrically, the associated linear transformation collapses area or volume to zero, so an inverse cannot exist.