How do I prove or disprove this matrix identity?

Let us have two n×n matrices A and B with real entries. Either prove, or disprove by providing a counterexample, that if ABA=0 and B is invertible, then A²=0.

Here is a 2×2 counterexample, easily extendable to n×n: Let A orthogonally project onto one axis, and let B rotate the plane by 90∘. The operation of ABA is to collapse everything down to one axis, then turn that axis, then collapse that axis down to the origin. However, A2=A≠0

Algebra

2 years ago

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Math Calculus — AID—

Total time spent: 4 hours 15 minutes (it is equivalent to 255 minutes). Let the flight time from Paris to Glasgow be X. Then the flight time from Glasgow to Paris will be (X+10). We have an equation with one unknown variable: X + 45 + (X+10) = 255 ⇒ X = 100 So the […]

What were the typical crafts for Maya?

Mayans are well-known as great pottery makers. They did not use the pottery wheel. However,the tecnique they used was coiling consecutive rings of clay on top of each other and smoothing the surface by hand. Also, they were good when it came to working with stone – stone craving is another important craft of Maya. Many […]

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Cigarette smoking causes emphysema.

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