SOLUTION: a square matrix A is said to be idempotent A^2=A (a)give an example of idempotent matrix other than 0 and I (b)show that,if matrix A is both idempotent and invertible then A=I

Algebra ->  Matrices-and-determiminant -> SOLUTION: a square matrix A is said to be idempotent A^2=A (a)give an example of idempotent matrix other than 0 and I (b)show that,if matrix A is both idempotent and invertible then A=I      Log On


   

Question 1062852: a square matrix A is said to be idempotent A^2=A
(a)give an example of idempotent matrix other than 0 and I
(b)show that,if matrix A is both idempotent and invertible then A=I
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Just the (a) part:

Let A = %28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29

be idempotent.  Then

   A%5E2%22%22=%22%22A

 %28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29%5E2%22%22=%22%22%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29

%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29%2A%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29%22%22=%22%22%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29

%28matrix%282%2C2%2Cw%5E2%2Bxy%2Cwx%2Bxz%2Cwy%2Byz%2Cxy%2Bz%5E2%29%29%22%22=%22%22%28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29

So we set each elements on the left equal to the corresponding
element on the right:

system%28w%5E2%2Bxy=w%2C+wx%2Bxz=x%2C+wy%2Byz=y%2C+xy%2Bz%5E2=z%29

Divide the second one through by x or the third one through
by y:

w + z = 1 

So we pick, say w = 1/4 and z = 3/4 since they have sum 1 

Substituting in

w%5E2%2Bxy=w

%281%2F4%29%5E2%2Bxy=1%2F4

1%2F16%2Bxy=1%2F4

1%2B16xy=4
 16xy=3
 xy=3%2F16

Choose x=1 and y=3/16, because their product is 3/16.

Substitute in

A = %28matrix%282%2C2%2Cw%2Cx%2Cy%2Cz%29%29

A = %28matrix%282%2C4%2C1%2F4%2C%22%22%2C%22%22%2C1%2C3%2F16%2C%22%22%2C%22%22%2C3%2F4%29%29%29

Edwin