Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 2 / b

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- Solution b

Solution

False in general. For symmetric matrices,

(A B)_{ji} = A_{jk} B_{ki} = B_{ik} A_{kj} = (B A)_{ij}

. The symmetric product criterion for symmetric matrices is therefore

A B = B A

. For example,

A = (1000), B = (0110), A B = (0010)

has

(A B)_{12} = 1

and

(A B)_{21} = 0

despite both factors being symmetric.

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