determinant cofactor expansion
A 2x2 determinant was an area. When the matrix grows to 3x3 and 4x4 the picture does not change kind. The dimension goes up by one. A 3x3 determinant is the volume of a parallelepiped, and an n x n determinant is a signed n-dimensional volume. Computing a large determinant is simple. Walk along one row and split it into a signed sum of smaller determinants. Add, subtract, add, the checkerboard of signs. That is cofactor expansion. From this one tool come Cramer's rule, why a determinant is multilinear and alternating, and why it is 0 when two columns are the same.
Unfold a 3x3 determinant directly. Walk along the first row one cell at a time with the buttons. Choose the j-th cell and that term turns on, and the row and column of that term are erased. The 2x2 that remains is the minor of that term, and that small determinant is the area we already know. The signs alternate, plus then minus then plus. So the whole determinant is a11 times M11, minus a12 times M12, plus a13 times M13. The running sum grows one step at a time and lands on det at the end. A large determinant splits into a signed sum of smaller ones. For a 4x4 you do the same thing one layer deeper.
A cofactor has two parts, a minor and a sign. Press any cell aij on the grid and that cell is highlighted, and the 2x2 minor Mij appears after row i and column j are erased. Beside it the sign (-1)i+j is shown. If the row number plus the column number is even the sign is plus, and if it is odd the sign is minus. So the signs lie down like a checkerboard from the top left. The minor times this sign is the cofactor Cij = (-1)i+j Mij. The expansion in the previous picture is the sum, along the first row, of aij times Cij. Expand along any row or any column and you get the same det. The sign board guarantees that.
Now solve a linear system with determinants. In A x = b the matrix A is a fixed invertible matrix, and the sliders set the right-hand side b1, b2. Cramer's rule gives the answer as a ratio of determinants. x1 is the determinant of the matrix with column 1 of A replaced by b, divided by det(A), and x2 is the determinant with column 2 replaced by b, divided by det(A). The two replaced matrices are shown as parallelogram areas, so how many times the numerator area fits in the denominator area is x. Move b and those parallelograms grow or shrink, and x1, x2 follow. Because a determinant is a volume, the answer of a linear system comes out as a ratio of volumes. When det(A) nears 0 the denominator vanishes and the answer blows up.
Feel the sign of a determinant with the slider. Move a 2x2 matrix and the parallelogram that is the image of the unit square moves with it. The area of that parallelogram is the absolute value |det|. The sign is direction. If the turn from the first axis to the second is counter-clockwise, det is positive. If it is clockwise, det is negative. Ease the slider across the moment when the two arrows reverse their turn, and at that moment |det| passes through 0 and the sign changes from plus to minus. In 2 dimensions it is a signed area, in 3 dimensions a signed volume, and in n dimensions a signed n-dimensional volume. The size is how much it grew and the sign is whether it flipped, and one determinant holds both.
A determinant has two characters. It is multilinear and it is alternating. Swap the two columns with the button and the sign flips. The size stays and plus becomes minus, and that is alternating. Turn one column toward the other with the slider and the columns become parallel, and at the moment they overlap completely the parallelogram collapses to a line and det becomes 0. If two columns are equal the determinant is 0, a plain consequence of being alternating. Swapping the same column with itself should flip the sign, but swapping a thing with itself should change nothing, so det = -det, which means det = 0. In volume it is easier. If two edges lie along one direction the shape is flat and has no volume.