A square matrix A is invertible if there exists A^{-1} such that AA^{-1} = A^{-1}A = I. A matrix is invertible if and only if it has full rank and non-zero determinant. Invertibility is equivalent to being non-singular and having linearly independent rows and columns.
From your study of matrix multiplication, you know that multiplying by a matrix transforms vectors — it can rotate, scale, shear, or project them. The matrix inverse is the transformation that undoes this: if A sends vector x to Ax, then A⁻¹ brings it back. The defining condition AA⁻¹ = A⁻¹A = I captures exactly this idea, where the identity matrix I is the "do nothing" transformation. Just as the number 1 satisfies a·(1/a) = 1, the inverse matrix satisfies the matrix analog of this equation.
Why does invertibility require the matrix to be square? A non-square matrix maps between spaces of different dimensions — for example, a 2×3 matrix maps ℝ³ into ℝ². No matter how you try to undo it, you can't recover the original three-dimensional information from two-dimensional output. Squareness is necessary for even the possibility of a two-sided inverse. But squareness isn't sufficient: a square matrix that collapses multiple input vectors to the same output (like a projection matrix) cannot be inverted, because you can't tell which input produced which output.
The determinant gives a scalar test for this collapse: det(A) = 0 if and only if A fails to be invertible. Geometrically, the determinant measures the factor by which A scales area (in 2D) or volume (in 3D). A determinant of zero means the transformation squashes space down to a lower dimension — exactly the unrecoverable collapse. Full rank — meaning all rows and all columns are linearly independent — is the equivalent algebraic condition: each row and column contributes genuinely new information.
The inverse matters most for solving systems of equations. If Ax = b has a unique solution, it's x = A⁻¹b. In practice, computing A⁻¹ explicitly is expensive and numerically unstable, so algorithms like Gaussian elimination solve Ax = b directly. But the existence of A⁻¹ is what guarantees a unique solution exists in the first place — making invertibility one of the most important properties a matrix can have.