The determinant decides whether an inverse exists
Determinant, trace, transposed matrix and an invertibility check for 2×2 and 3×3 matrices.
The determinant is the defining number of a square matrix. Geometrically it is the factor by which the transformation scales area or volume, and its sign says whether orientation flips.
The practical reading is simpler: a zero determinant means no inverse exists and a system with that matrix has no unique solution. Which is why it is the first thing anyone computes.
What this calculator shows
- The determinant and the trace
- Whether an inverse matrix exists
- The transposed matrix and the rank
What to keep in mind
- Square 2×2 and 3×3 matrices are supported; the 3×3 determinant expands along the first row.
- Enter numbers row by row, spaces between them and a new line per row — the way you would write it out.
FAQs
What does a zero determinant mean?
Rows or columns are linearly dependent, no inverse exists, and the transformation collapses space onto a plane or line. Systems with such a matrix have no unique solution.
How is a 3×3 determinant computed?
By expansion along the first row: each element times the determinant of the remaining 2×2, with alternating signs. Sarrus's rule gives the same answer.
What is the trace?
The sum of the main-diagonal entries. It equals the sum of the eigenvalues and is invariant under a change of basis, which makes it a genuinely useful quantity.
Why transpose a matrix?
It swaps rows and columns — needed in statistics for data matrices, in graphics for transforming normals, and simply to test whether a matrix is symmetric.
Worked example
A 2×2 matrix
Input: 4 7 / 2 6
Output: Determinant 10, trace 10
Note: The determinant is non-zero, so an inverse exists. Determinant and trace matching here is coincidence — usually they differ.