Undoing a Matrix

Just as division undoes multiplication, the matrix inverse undoes a linear transformation. If A maps x to b, then A⁻¹ maps b back to x — the reason inverse matrices solve systems of equations in one step.

The Adjugate Formula

A⁻¹ = adj(A) / det(A). Build the cofactor matrix by computing each element's minor with alternating signs, transpose it to get the adjugate, and divide every entry by the determinant. For 2×2 matrices this collapses to the familiar swap-and-negate shortcut.

When It Fails

A zero determinant means no inverse exists — the transformation squashes space onto a lower dimension, so information is lost and nothing can undo it. The calculator flags this case and shows the identity-matrix verification whenever an inverse does exist.