What is $A^{-1}$ for $A = \begin{bmatrix}1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 4 & 1\end{bmatrix}$?

Explanation

1. 01

   $|A| = 1$ (a triangular matrix: the product of the diagonal), so $A^{-1} = \operatorname{adj} A$.

2. 02

   The cofactors of row 3 are $A_{31} = 0$, $A_{32} = 0$, $A_{33} = 1$; of row 2, $A_{21} = 0$, $A_{22} = 1$, $A_{23} = -4$; of row 1, $A_{11} = 1$, $A_{12} = -2$, $A_{13} = 5$.

3. 03

   Transposing the cofactor matrix gives $A^{-1} = \begin{bmatrix}1 & 0 & 0 \\ -2 & 1 & 0 \\ 5 & -4 & 1\end{bmatrix}$.