Use a determinant to check a two-variable transformation

Define the task

Write a two-variable linear transformation as a matrix with one output variable per row and one input variable per column. Record the variable order. In two dimensions, the determinant describes signed area scaling. A negative sign reverses orientation, and a zero value indicates that the transformation collapses the plane to a lower-dimensional set.

Carry out the check

For entries a,b in the first row and c,d in the second, compute ad−bc. Check both products independently when they are nearly equal. Floating-point cancellation can make a small determinant difficult to interpret. For larger matrices use a numerical library and consider a logarithmic determinant to avoid overflow.

Worked hypothetical example

For a hypothetical matrix with rows (2,1) and (0,3), a unit square maps to a parallelogram with area six because the determinant is 6. Exchanging the rows changes the determinant to −6 while preserving the area magnitude. A tiny determinant alone is not a scale-independent test for instability. A diagonal matrix with both entries 0.001 has determinant 0.000001 but equal singular values and condition number one.

Method checked 12 November 2026 against R documentation: Calculate the Determinant of a Matrix.

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