Question

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

Answer 1

det(cA) = $c^n$det(A), where $n$ is the dimension of the matrix (n rows, n columns).

det($A^T$) = det(A) since transposing a matrix doesn't change its determinant.

So $det(3A^T)$ = $3^ndet(A^T)=3^ndet(A)=3^n \cdot 2=18, \, 3^n=9,$ $\,n=2.$

Answer: the order of matrix $A$ is 2.