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Linear Algebra : Zero Matrix Proof

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Let A be a 2 x 2 matrix with A^3 = O. Prove that A^2 = O. Where O is the zero matrix. (Note: I've said nothing about the invertibility of A, it may or may not be invertible).

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Solution Summary

A proof is offered for an equality to the zero matrix. The invertibility of a matrix function is given.

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*If A is invertible, i.e. |A|<>0
then we have: (A)^-1 * A*A*A = (A)^-1 * 0
so A*A=0

*If A isn't invertible, so |A|=0
i.e. a11*a22 - a12*a21=0
or a11*a22 = a12*a21

then A*A=
( a11^2+a12*a21 a11*a12+a12*a22
a11*a21 +a21*a22 a22^2*a12*a21)

And let A*A*A=
(B11 ...

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