$matrix a^2$

For instance, if we assume b is 0 we can use the equations to find one example of an idempotent matrix with 0 in the top right. Obviously using four variables is really unwieldy ,so you want to do something to clean it up.Īssuming you don't need to find every such matrix, just one, a trick you can do is try setting some of these variables equal to 0. One way is by doing what you've done and just playing around until you've found something. The first thing I'll point out is that if you only have to find one such example, the identity matrix clearly works. So there are a few ways you could go about trying to solve this.

If a 2=1, then you know a = 1 (and not -1) from a (ad-bc) = (ad-bc), above. If you look at your determinant equation in that light, you get a 2=. If you backtrack, and say a+d is not 1, then you get a=d (see 'Third' above), b=0, and c=0. Which means you're done with that case - you have two degrees of freedom, a and either b or c. If you plug that into equations 1 and 4, you should see that they are satisfied equations 2 and 3 were already satisfied with a + d = 1. If you assume a+d = 1, and plug that into (EDIT(clarity): your equation 4), you get bc + (1-a) 2 = 1-a, which simplifies to bc + a(a-1) = 0 since a-1=-d, this is actually bc-ad=0 for that case, which means that if a+d = 1, the determinant must be 0, e.g. Which turns into either the determinant is 0 (which we already had as a possibility) or a = 1, which is exciting. This makes the resulting right side equal to the determinant: Now, some more clever ideas: take equation 1 times d minus equation 2 times c. Similarly, if you add equations 2 and 3 together, you get (a+d)(b+c)=(b+c), which, once again, gives you an alternative b+c=0 case (EDIT: This is redundant with the b=c=0 cases, though, so kinda not useful) (a-d)(a+d)=(a-d) - I bet you did this - but you discarded the a=d case, which you should also analyze. Third: Subtract eqaution 4 from equation 1. Second: what about the determinant? Since det(A) 2 =det(A), that should give you det(A) = 0 or 1, e.g. Just ways to keep pushing stuff forward.įirst, a nitpick: from the above equations, you can also get c = 0 or b = 0. Ok, so, here's some random looks at the equations, without trying to change your basic approach or anything.

Matrix a^2