Adjugate Matrix/Examples/Arbitrary Matrix 3

Example of Adjugate Matrix
Let $\mathbf A$ be the square matrix:


 * $\mathbf A = \begin {pmatrix} 1 & 2 & 0 \\ 0 & -1 & 2 \\ -1 & 2 & 0 \end {pmatrix}$

Then the adjugate matrix of $\mathbf A$ is:


 * $\adj {\mathbf A} = \begin {pmatrix} -4 & 0 & 4 \\ -2 & 0 & -2 \\ -1 & -4 & -1 \end {pmatrix}$

Proof
For a square matrix $\mathbf A = a_{i j}$ of order $3$, the adjugate matrix of $\mathbf A$ is:


 * $\adj {\mathbf A} = \begin {pmatrix} A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33} \end {pmatrix}$

For each $a_{i j}$ in $\mathbf A$, we calculate the cofactors $A_{i j}$:

Hence:
 * $\adj {\mathbf A} = \begin {pmatrix} -4 & 0 & 4 \\ -2 & 0 & -2 \\ -1 & -4 & -1 \end {pmatrix}$

We check this result, using Product of Matrix with Adjugate equals Determinant by Unit Matrix.

First we note that $\map \det {\mathbf A} = -8$, by Expansion Theorem for Determinants, for example expanding using column $1$.

and all is well.