The Determinant Grows into Volume
Remember the 2x2 determinant was area. So what happens when the matrix grows to 3x3, 4x4? The picture doesn't get replaced; the dimension just climbs by one. A 3x3 is the volume of a parallelepiped, and an n by n is signed n-dimensional volume. The catch is computing a big determinant, and the answer is surprisingly plain: walk along one row and split it into a signed sum of smaller determinants. Add, subtract, add, with a +,-,+ checkerboard of signs. That is cofactor expansion. From this one tool flow Cramer's rule, why the determinant is multilinear and alternating, and why two equal columns force it to zero.
Let's unfold a 3x3 determinant by hand. The buttons walk you across the top row, one cell at a time. Pick cell j and that entry lights up, then its whole row and column get crossed out. What's left is a 2x2, the minor of that entry, and that little determinant is the area we already know. Each cell carries an alternating +,-,+ sign. So the full determinant is a11 times M11 minus a12 times M12 plus a13 times M13. Step through and the running sum grows until it lands exactly on det. The key is that a big determinant breaks into a signed sum of small ones. For a 4x4 you just do the same thing one layer deeper.
A cofactor has two parts: a minor and a sign. Click any cell aij in the grid. That cell highlights, and the 2x2 minor Mij left after deleting its row i and column j appears. Beside it sits the sign (-1)i+j: plus when row number plus column number is even, minus when odd. So the signs tile like a checkerboard from the top left, +,-,+,-. The minor times this sign is the cofactor Cij = (-1)i+j Mij. The expansion in the last widget was just the sum of aij times Cij across the top row. Expand along any row or any column and you get the same det. The sign board is what guarantees that.
Now solve a system with determinants. In A x = b, A is a fixed invertible matrix and the sliders set the right side b1, b2. Cramer's rule gives the answer as a ratio of determinants: x1 is the determinant of A with column 1 replaced by b, divided by det(A); x2 is the determinant of A with column 2 replaced by b, over det(A). The two replaced matrices show up as parallelogram areas. How many times the numerator area is the denominator area is exactly x. Move b and that parallelogram grows or shrinks, and x1, x2 follow. Because a determinant is volume, the system's answer comes out as a ratio of volumes, and that is the spirit of Cramer. When det(A) nears 0 the denominator vanishes and the answer blows up, which you will see too.
Feel what the sign of a determinant means by hand. Move the 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 of the determinant, |det|. The sign is orientation: if the turn from î to ĵ goes counterclockwise, det is positive; clockwise, negative. Ease the sliders until the rotation of the two arrows flips, and watch that moment: |det| passes through 0 and the sign switches from + to -. In two dimensions it is signed area, in three signed volume, in n dimensions signed n-volume. Size is how much it grew, sign is whether it flipped. One determinant carries both.
Last, the two characters of the determinant: it is multilinear and alternating. Toggle a swap of two columns. The sign just flips. The det is the same size but + becomes -. That is the alternating property. Then use the slider to rotate one column toward the other. The two columns line up more and more, and at the instant they fully overlap the parallelogram collapses into a single line and det becomes 0. Two equal columns force the determinant to 0, a plain consequence of alternation: swapping the two equal columns should flip the sign, yet swapping identical things changes nothing, so det = -det, which means det = 0. As volume it is even easier: when two edges coincide in one direction the shape flattens and has no volume.