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.
The determinant and the trace Whether an inverse matrix exists The transposed matrix and the rank 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. 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.
