You can't divide by a matrix, but if it has an inverse, you can multiply by that instead. The inverse "undoes" the original matrix like a reciprocal undoes a number. Here A = [[2, 1], [1, d]]. Change d, the bottom-right value, and watch A, its inverse and the determinant ad − bc change, right up to the point where the inverse disappears.
A turns the white unit square (area 1) into the shaded shape. Its area is the size of the determinant: 7.00 square units.
For a 2×2 matrix A = [[a,b],[c,d]], the inverse is A⁻¹ = (1/det(A))·[[d,−b],[−c,a]]. Notice the determinant sits in the denominator — when det(A) = ad − bc = 0, division by zero makes the inverse undefined. The formula swaps the diagonal elements, negates the off-diagonal ones, then scales by 1/det(A). Only square matrices can have inverses, and only when their determinant is non-zero. A 3×3 or larger matrix uses a more complex formula, but the principle is identical: det(A) ≠ 0 is the gate.
The inverse of a matrix acts like a reciprocal: A·A⁻¹ = I, where I is the identity matrix, but it only exists for square matrices with non-zero determinant.