Find if $det(A)=det(A^n)$ for $n>1$.
How do I tackle questions like this, in general if the matrix $A$ is provided in the question? Should I work out with the basic definition of a determinant, which I found very difficult to apply in any question? I cannot of course go on to calculate the value of $det(A^n)$?
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Let $\mathbf{A}$ and $\mathbf{B}$ be matrices. Then it is well known that $\det$ is linear w.r.t matrix multiplication, i.e $$\det(\mathbf{A}\mathbf{B})=\det(\mathbf{A})\det(\mathbf{B})$$ Therefore $$\det(\mathbf{A}^n)=\det\left(\prod_{i=1}^{n}\mathbf{A}\right)$$ $$=\prod_{i=1}^{n}\det(\mathbf{A})=(\det(\mathbf{A}))^n.$$
The determinant function is multiplicative, i.e. $\det(AB) = \det(A)\det(B)$. In particular, this means that $\det(A^n) = \det(A)^n$. So given a matrix $A$, just find its determinant and raise it to the given power.
For instance, let
$$A = \left[ \begin{matrix} 6 & 2 \\ 7 & 5 \end{matrix} \right]$$
Then $\det(A) = 6 \cdot 5 - 2 \cdot 7 = 16 = 2^4$. So if I asked you to find $\det(A^n)$ you could just tell me $16^n$ or $2^{4n}$, whichever fits your fancy.
In fact, I could even specify large $n$ and you might calculate it easily enough, if you're familiar enough with your powers of $2$ in this case. For instance, $\det(A^4) = 2^{16} = 65536$.