How this instrument works
For a 2×2 matrix [[a,b],[c,d]], the determinant, ad−bc, measures how the matrix scales area: a determinant of zero means the matrix squashes the plane down to a line (or a point), losing a dimension, and a nonzero determinant means the matrix is invertible, with its magnitude showing exactly how much area scales up or down under the transformation. The trace, a+d, is simply the sum of the diagonal entries.
Both numbers connect directly to the matrix's eigenvalues, even without solving for those eigenvalues explicitly: the trace always equals the SUM of the two eigenvalues, and the determinant always equals their PRODUCT — a genuine shortcut, since these two simple sums and products can be read straight off the original matrix.
This page is deliberately scoped to 2×2 matrices, where both quantities reduce to simple, closed-form expressions in the four entries. Larger matrices need more elaborate methods (cofactor expansion for the determinant, still just a diagonal sum for the trace), handled by dedicated linear-algebra tools rather than this quick 2×2-specific page.
- Enter the matrix's four entries into the a, b, c, and d fields, reading left to right, top to bottom.
- Read Determinant and Trace: the sheet applies ad−bc and a+d directly.
- A determinant of exactly zero flags a singular, non-invertible matrix.
Worked example — [[2,1],[3,4]]
The matrix [[2,1],[3,4]] has a determinant of 2×4−1×3=5 and a trace of 2+4=6 — a nonzero determinant confirms this matrix is invertible, and the trace, 6, equals the sum of its two eigenvalues, whatever those turn out to be individually.
The identity matrix [[1,0],[0,1]] has a determinant of exactly 1 (it doesn't scale area at all) and a trace of 2. The matrix [[2,4],[1,2]], by contrast, has a determinant of 2×2−4×1=0 — singular, non-invertible, since its two rows are proportional (row 2 is exactly half of row 1).
Questions
What does a matrix's determinant tell you?
It measures how the matrix scales area under its own transformation. A nonzero determinant means the matrix is invertible; a zero determinant means it squashes the plane down to a lower dimension, a line or a point.
What is the trace of a matrix?
The sum of its diagonal entries — for a 2×2 matrix [[a,b],[c,d]], that's simply a+d, ignoring the off-diagonal entries b and c entirely.
How are determinant and trace related to eigenvalues?
For any square matrix, the trace equals the sum of all its eigenvalues, and the determinant equals their product — a shortcut that reveals those two combined facts about the eigenvalues without solving for them individually.
What does a determinant of zero mean?
The matrix is singular (non-invertible) — its rows (or columns) are linearly dependent, meaning one is some multiple or combination of the other, and the matrix collapses the plane down to a line or a single point.
Why is this page limited to 2×2 matrices?
Because both the determinant and trace reduce to simple, direct formulas in the four entries for a 2×2 matrix specifically. Larger matrices need more elaborate methods (like cofactor expansion) for the determinant, handled by other dedicated tools.