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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 6A / b / Solution

If

A_{ji}

denotes the matrix obtained by deleting row

j

and column

i

, then the adjugate matrix is

adj (A)_{ij} = (- 1)^{i + j} det A_{ji} .

For nonsingular

A

,

adj (A) = det (A) A^{- 1} .

Therefore, when

A

and

B

are nonsingular,

adj (A B) = det (A) det (B) (A B)^{- 1} = det (B) B^{- 1} det (A) A^{- 1} = adj (B) adj (A) .

Solved by gpt-5.6-sol high.

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