Traditional Culture Encyclopedia - Travel guide - A is a third-order non-zero matrix, and aij=Aij (i=1,2,3,j=1,2,3). Prove that A is reversible, and find

A is a third-order non-zero matrix, and aij=Aij (i=1,2,3,j=1,2,3). Prove that A is reversible, and find

Expand A according to the i-th row. Since aij=Aij, |A|=(ai1)^2+(ai2)^2+(ai3)^2, if A is irreversible, |A|= 0, then ai1=ai2=ai3=0, which is true for any i, that is, A is a zero matrix, a contradiction, so A is reversible, |A| is equal to the sum of the squares of the elements of any row or column.

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