Answer to Question #222613 in Linear Algebra for Anuj

Question

Given a number 𝑑 = 𝑑𝑒𝑡(𝐴1, ⋯ , 𝐴𝑛 ), find 𝑑𝑒𝑡(𝐴𝑛, ⋯ , 𝐴1 ).

Answer 1

Property of determinant

$$det(A_1, ..., A_i, ..., A_j, ..., A_n)=-det(A_1, ..., A_j, ..., A_i ..., A_n)$$

We swap $A_1$ and $A_n,$ $A_2$ and $A_{n-1},...$

If $n=4k-3$ or $n=4k,k=1,2,...,$ then

$$det(A_1, ..., A_n)=det(A_n,...A_1)$$

If $n=4k-2$ or $n=4k-1,k=1,2,...,$ then

$$det(A_1, ..., A_n)=-det(A_n,...A_1)$$