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Since A has characteristic polynomial x(x+1)²(x-1)³, its eigenvalues are 0, -1, -1, 1, 1, 1.

Since A is diagonal, we know that A has its diagonal entries 0, -1, -1, 1, 1, 1 in some order.

Suppose that A has diagonal entries in this order (if not, the algebra below yields a similar set of equations otherwise).

Questions for you:

What happens if you multiply a generic 6 x 6 matrix to the left of the diagonal matrix A?

What happens if you multiply a generic 6 x 6 matrix to the right of the diagonal matrix A?

(Block multiplication with 3 x 3 blocks for both A and B can expedite matters; A is then block diagonal whose lower diagonal block is the 3 x 3 identity matrix.)

Solve the system, and see how many free variables there are. That will be the desired dimension.

I hope this helps!

God positioned human beings in this earth for one purpose. His unique purpose became glaring whilst He instructed Adam and Eve to "be fruitful and alter into many and fill the earth and subdue it, and have in subjection the fish of the sea and the flying creatures of the heavens and each residing creature it is shifting upon the earth" Genesis a million:28 His purpose became for persons to stay in the international continuously, perfect human beings. yet of path, Eve and Adam sinned, brought about the downfall of guy. This altered God's unique purpose for humankind.

Sorry no cheating HERE ! Google it !