Exercise 9, § 11

Algebra, III, 648

The adjoint of a square matrix $X$ of order $n$ over A is the matrix $\tilde{X} = (\det(X^{ji}))$ of minors of $X$ of order $n - 1$ . Show that if $X$ is invertible then $\det(\tilde{X}) = (\det X)^{n-1}$ and that every minor $\det(\tilde{X}_{H', K'})$ of $\tilde{X}$ of order $p$ is given by the formula

\det(\tilde{X}_{H', K'}) = (\det X)^{p-1}(X_{H'', K''})

where

H'

and

K'

are the complements of

H

and

K

respectively in

\{1, n\}

("Jacobi's identities"; use relation (80) of no. 11).

Solution

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