2.2.2—Determinants and inverses

For an invertible square matrix A, A⁻¹=adj(A)/det(A). The determinant must be non-zero; for a 2×2 matrix [[a,b],[c,d]], the inverse is (1/(ad−bc))[[d,−b],[−c,a]].

The adjugate is the transpose of the cofactor matrix. Use a determinant test first, then verify the inverse by multiplication or by checking the formula’s dimensions.

For [[2,1],[1,1]], det=1 and the inverse is [[1,−1],[−1,2]]. Multiplying the two matrices gives the identity.

Changing one sign in the 2×2 formula can invalidate every entry, and det=0 means the formula is undefined rather than an inverse with infinite values.