det(A^4)=0

**ssl000** · December 22nd 2008, 08:34 AM

Suppose that A is a square matrix such that det A^4=0. Explain why A cannot be invertible.

**Isomorphism** · December 22nd 2008, 09:05 AM

Originally Posted by ssl000

Suppose that A is a square matrix such that det A^4=0. Explain why A cannot be invertible.

Hint: $\text{det }A^4 = (\text{det }A)^4$

**ssl000** · December 22nd 2008, 01:52 PM

I know of the theorem, det(AB)=det(A)*det(B). det(A^4)=det(A)*det(A)*det(A)*det(A) will equal to zero, making it invertible, but I still don't understand why it will equal to zero.

**TheMasterMind** · December 22nd 2008, 02:20 PM

Originally Posted by ssl000

Suppose that A is a square matrix such that det A^4=0. Explain why A cannot be invertible.

$det A^4=0$ means $(det A)^4=0$ so $det A=0$ and therefore $A$ is not invertible.

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