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. Watch what happens when you change the bottom-right value and cross the point where the matrix loses its inverse.
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.