anonymous
  • anonymous
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?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
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
anonymous
  • anonymous
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
anonymous
  • anonymous
proof:

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anonymous
  • anonymous
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

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