16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 5.4.1

The classical adjoint formula states that $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$, where the adjugate matrix is the transpose of the cofactor matrix. To find each inverse, you'll compute the determinant first (to check invertibility), then systematically find all nine cofactors by computing $2 \times 2$ minors with appropriate signs, transpose that cofactor matrix, and scale by $1/\det(A)$. The second matrix is a rotation, so expect its inverse to have a clean geometric form—notice how the structure suggests what the answer should look like before you compute.

The classical adjoint formula states that $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$, where the adjugate matrix is the transpose of the cofactor matrix. To find each inverse, you'll compute the determinant first (to check invertibility), then systematically find all nine cofactors by computing $2 \times 2$ minors with appropriate signs, transpose that cofactor matrix, and scale by $1/\det(A)$. The second matrix is a rotation, so expect its inverse to have a clean geometric form—notice how the structure suggests what the answer should look like before you compute.