Value of Adjugate of Determinant

Theorem

Let $D$ be the determinant of order $n$.

Let $D^*$ be the adjugate of $D$.

Then $D^* = D^{n - 1}$.

Proof

Let $\mathbf A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\

 a_{21} & a_{22} & \cdots & a_{2n} \\

\vdots & \vdots & \ddots & \vdots \\

 a_{n1} & a_{n2} & \cdots & a_{nn}\end{bmatrix}$ and $\mathbf A^* = \begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1n} \\
 A_{21} & A_{22} & \cdots & A_{2n} \\

\vdots & \vdots & \ddots & \vdots \\

 A_{n1} & A_{n2} & \cdots & A_{nn}\end{bmatrix}$.

Thus:

$\paren {\mathbf A^*}^\intercal = \begin{bmatrix} A_{11} & A_{21} & \cdots & A_{n1} \\

 A_{12} & A_{22} & \cdots & A_{n2} \\

\vdots & \vdots & \ddots & \vdots \\

 A_{1n} & A_{2n} & \cdots & A_{nn}\end{bmatrix}$

is the transpose of $\mathbf A^*$.

Let $c_{ij}$ be the typical element of $\mathbf A \paren {\mathbf A^*}^\intercal$.

Then by definition of matrix product:

$\ds c_{ij} = \sum_{k \mathop = 1}^n a_{ik} A_{jk}$

Thus by the corollary of the Expansion Theorem for Determinants:

$c_{ij} = \delta_{ij} D$

So by Determinant of Diagonal Matrix:

$\map \det {\mathbf A \paren {\mathbf A^*}^\intercal} = \begin{vmatrix} D & 0 & \cdots & 0 \\

 0 & D & \cdots & 0 \\

\vdots & \vdots & \ddots & \vdots \\

 0 & 0 & \cdots & D\end{vmatrix} = D^n$

From Determinant of Matrix Product:

$\map \det {\mathbf A} \map \det {\paren {\mathbf A^*}^\intercal} = \map \det {\mathbf A \paren {\mathbf A^*}^\intercal}$

From Determinant of Transpose:

$\map \det {\paren {\mathbf A^*}^\intercal} = \map \det {\mathbf A^*}$

Thus as $D = \map \det {\mathbf A}$ and $D^* = \map \det {\mathbf A^*}$ it follows that:

$DD^* = D^n$

Now if $D \ne 0$, the result follows.

However, if $D = 0$ we need to show that $D^* = 0$.

Let $D^* = \begin{vmatrix} A_{11} & A_{12} & \cdots & A_{1n} \\

 A_{21} & A_{22} & \cdots & A_{2n} \\

\vdots & \vdots & \ddots & \vdots \\

 A_{n1} & A_{n2} & \cdots & A_{nn}\end{vmatrix}$.

Suppose that at least one element of $\mathbf A$, say $a_{rs}$, is non-zero (otherwise the result follows immediately).

By Expansion Theorem for Determinants and its corollary, we can expand $D$ by row $r$, and get:

$\ds D = 0 = \sum_{j \mathop = 1}^n A_{ij} t_j, \forall i = 1, 2, \ldots, n$

for all $t_1 = a_{r1}, t_2 = a_{r2}, \ldots, t_n = a_{rn}$.

But $t_s = a_{rs} \ne 0$.

So, by (work in progress):

$D^* = \begin{vmatrix} A_{11} & A_{12} & \cdots & A_{1n} \\

 A_{21} & A_{22} & \cdots & A_{2n} \\

\vdots & \vdots & \ddots & \vdots \\

 A_{n1} & A_{n2} & \cdots & A_{nn}\end{vmatrix} = 0$

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