Question

If A is 3×3 invertible matrix, then what will be the value of k, if
det(A^-1) = (detA)^k ?

Answer 1

det(A^-1)= (det(A))^(-1)

Now lets have a look a the reason of it,

A* A^-1 =I

taking determinant both side we get,

det(A* A^-1 )=det(I)=1

Also,

det(A*B)=det(A)*det(B)

Hence we get,

det(A)=det(A^-1) ^-1

OR

det(A^-1)= (det(A))^(-1)
So k=-1