Question

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

Answer 1

Suppose $A \in M_n(\mathbb R).$

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 $t \in \mathbb Z$ because $A^t$ might not be defined otherwise.