4×4 Matrix Determinant Calculator

How it is calculated

The tool reads the 16 entries a11 to a44 (row first, then column) and reduces the matrix to upper-triangular form by Gaussian elimination with partial pivoting. In each column it takes the row with the largest absolute value on or below the diagonal, swaps it up (every swap flips the sign), multiplies the running determinant by that pivot and subtracts multiples of the pivot row from the rows below. The result is the signed product of the four pivots: det = ± p1 × p2 × p3 × p4. Tolerances are relative to the largest entry, so a matrix scaled by 0.001 (determinant 1e-12) is still reported as invertible. Values below 0.00005 are printed in scientific notation instead of 0.0000. Singular means |det| < 1e-9 × (largest entry)⁴. All 16 cells must hold numbers.

Common mistakes

• Typing a row in the wrong position or swapping two rows by accident. A transposed matrix has the same determinant, but one swapped pair of rows flips the sign of the answer.
• Leaving a cell empty because it should be zero. All 16 cells need a value; a blank cell gives no result, so type 0.
• Reading a determinant such as 1.0000e-12 as "almost zero, so singular". For a matrix with entries around 0.001 that is the correct size (0.001⁴), and the tool judges singularity relative to the entries, so it says Invertible.
• Assuming that doubling every entry doubles the determinant. For a 4×4 matrix the determinant is multiplied by 2⁴ = 16.
• Entering a matrix whose row is a multiple of another, such as (1, 2, 3, 4) and (2, 4, 6, 8), and expecting a nonzero value. Such rows are linearly dependent and the tool correctly reports Singular with 0.0000.