Proof of Idempotent Matrix with Inverse = Identity Matrix

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SUMMARY

An idempotent matrix, defined by the property A² = A, can only have an inverse if it is the identity matrix. This conclusion arises from the fact that if A has an inverse (A⁻¹), multiplying both sides of the equation A² = A by A⁻¹ leads to the identity matrix. The discussion suggests that while one can explore arbitrary 3x3 matrices to understand this concept, the most elegant proof applies to all n x n matrices and confirms that the only idempotent matrix with an inverse is indeed the identity matrix.

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keelejody
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is there a direct proof that an idempotent matrix with inverse, can only be an identity matirx

i can't find about how id prove it

i know A^2=A and A^-1 exists

so too AB=BA

its obvious to say elements must be 1 or 0 but finding an overal rule isn't obvious to me
 
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If A has an inverse, multiply both sides of A2= A by A-1!
 
If you're feeling overly ambitious, you could also try setting up an arbitrary 3x3 matrix (like with entries a, b, c...). Multiply it by itself, and then set it equal to itself. You should come out with a system of equations that should end up proving that your arbitrary matrix is the identity.

The proof above (HallsofIvy) is much more elegant, and applicable for all nxn matrices, but setting up the arbitrary matrix would probably be a good way to practice your matrix math.
 

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