Question

Suppose A is a square matrix such that det(A0 = 2 and det(3A the power of t) = 18 then find the order of matrix A

Answer 1

Suppose A∈Mn​(R).

Then, 

$\det (3A^t) = 3^n\det(A^t) = 3^n\det(A^t) =3^n\det(A)=$

$=2 \cdot 3^n=18$

because the determinant is multilinear as a function of rows and the determinant respects the matrix multiplication. We have, then,

$3^n=9,$

which implies that  $n=2.$

The solution relies on the assumption that t∈Z because $A^t$ might not be defined otherwise.