If `alpha` is a characteristic root of a nonsin-gular matrix, then prove that `|A|/alpha` is a characteristic root of adj A.

Explanation

Since `alpha` is a characteristic root of a nonsingular matrix, therefore `alpha ne 0`. Also `alpha` is a characteristic root of A implies that there exists a nonzero vector X such that
`AX=alpha X`
or (adj. A) (AX) = (adj. A) `(alpha X)`
or `[(adj A)A] X=alpha` (adj A)X
or `|A|IX=alpha` (adj A) X`" "[ :' ("adj A")A=|A|I]`
or `|A|X=alpha`(adj A) X
or `(|A|)/alpha X=("adj A")X`
or `("adj A")X=(|A|)/alpha X`
Since X is nonzero vector, `|A//alpha|` is a characteristic root of the matrix adj A.