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How to find the determinant of a 4×4 matrix
There is no short formula for a 4×4 determinant. Written out in full it has 24 products of four entries each, so nobody computes it that way. There are two practical routes.
Cofactor expansion picks a row or a column and turns the problem into four 3×3 determinants, each multiplied by its entry and a sign from the checkerboard pattern + − + −. Every zero in the chosen row removes one 3×3 determinant, so always expand along the row or column with the most zeros.
Row reduction turns the matrix into an upper triangular one. Adding a multiple of one row to another does not change the determinant, swapping two rows changes its sign, and the determinant of a triangular matrix is the product of its diagonal. For a 4×4 matrix this is usually about a third of the arithmetic of a full expansion.
Worked example by row reduction
Take the matrix with rows (2, 1, 0, 1), (4, 3, 1, 2), (0, 2, 3, 1) and (2, 1, 1, 4).
Clear the first column. Row 2 minus 2 times row 1 gives (0, 1, 1, 0). Row 4 minus row 1 gives (0, 0, 1, 3). Row 3 already starts with a zero.
Clear the second column. Row 3 minus 2 times the new row 2 gives (0, 0, 1, 1). Then clear the third column: row 4 minus row 3 gives (0, 0, 0, 2).
The matrix is now upper triangular with diagonal 2, 1, 1, 2. No rows were swapped, so det(A) = 2·1·1·2 = 4.
Checking your answer
Both routes must give the same number, which makes one a check for the other. The calculator above is set to 4×4 and expands along a row or column by default, showing each 3×3 minor. Switch the method to triangular form to see the row reduction instead, or to Bareiss for a fraction-free elimination that keeps integer matrices in integers.