Question

So... I have to say if det(A^k) = det(A)^k is true or false and I have to prove why it's either true or false, can I get a little help?

Answer 1

we know that det ( A*B) = det ( A ) det B
so det (A^2) = det ( A*A)= det A * det A
det (A^3) = det ( A^2 * A) = det (A^2) * det A
which we know from before is det a * det a * det A

Answer 2

To make this pretty we can prove it by induction.
we already know that det( A * B) = det (A) * det B
so claim:
det ( A^k) = [ det A ] ^k

Answer 3

proof:

Answer 4

basis case, k = 2.
det ( A^2) = det (A*A) = det A * det A = [ det A ] ^2
this is true by theorem above.
Assume it is true for k
so det ( A^k) = [det A] ^k
now det (A^k+1) = det (A^k * A ) = det A^k det A = [ det A] ^k * det A = [ det A] ^k+1
And we have proven it for all k (or all n positive integers n>=2