Matrix Calculator

This calculator relies on foundational linear algebra principles to evaluate matrix characteristics. For a 2x2 matrix defined by rows (A1, A2) and (B1, B2), the determinant is calculated using the formula det2 = (A1 * B2) - (A2 * B1), and the trace is the sum of the main diagonal elements, tr2 = A1 + B2. For a 3x3 matrix adding a third row (C1, C2, C3), the determinant expands via Sarrus' rule or cofactor expansion into det3 = A1*B2*C3 + A2*B3*C1 + A3*B1*C2 - A3*B2*C1 - A2*B1*C3 - A1*B3*C2, while the trace sums all three diagonal entries as tr3 = A1 + B2 + C3.

Let us calculate the determinant and trace of a specific 3x3 matrix. Assume our matrix has the following elements: Row 1 (A1=2, A2=1, A3=3), Row 2 (B1=0, B2=4, B3=1), and Row 3 (C1=5, C2=2, C3=6). First, we find the trace by adding the main diagonal elements: tr3 = A1 + B2 + C3 = 2 + 4 + 6 = 12. Next, we compute the determinant by applying the 3x3 formula: det3 = (2 * 4 * 6) + (1 * 1 * 5) + (3 * 0 * 2) - (3 * 4 * 5) - (1 * 0 * 6) - (2 * 1 * 2). This simplifies to 48 + 5 + 0 - 60 - 0 - 4 = -11. Thus, the matrix has a trace of 12 and a determinant of -11.

Always double-check your input coordinates before running computations, as a single misplaced negative sign will completely alter your determinant and trace results. Keep in mind that traces are only defined for square matrices where the number of rows equals the number of columns. When working by hand alongside this tool, use structured grid paper to keep your rows and columns neatly aligned and avoid transposition errors.