# Thread: 2 Linear Algebra Question

1. ## 2 Linear Algebra Question

1.Let B an nxn matrix such that B^4=0

Show that A=I-B is invertible and A^-1 = I+B+B^2+B^3?

2 Let A an nxn Matrix
a.If A^2=0 show that A is not Invertible
b. If A^2= A and A is not the identity, show that A is not Invertible.

plz help.. i couldnt seem to figure it out?

2. Originally Posted by firebio

1.Let B an nxn matrix such that B^4=0

Show that A=I-B is invertible and A^-1 = I+B+B^2+B^3?
The definition of $\displaystyle A^{-1}$ is that $\displaystyle AA^{-1}= I$ and $\displaystyle A^{-1}A= I$

$\displaystyle (I- B)(I+ B+ B^2+ B^3)= I(I+ B+ B^2+ B^3)- B(I+ B+ B^2+ B^3)$$\displaystyle = (I+ B+ B^2+ B^3)-(B+ B^2+ B^3+ B^4)= I- B^4= I since \displaystyle B^4= 0. \displaystyle (I+ B+ B^2+ B^3)(I- B)= (I+ B+ B^2+ B^3)I- (I+ B+ B^2+ B^3)B$$\displaystyle = (I+ B+ B^2+ B^3)- (B+ B^2+ B^3+ B^4)= I- B^4= I$.

2 Let A an nxn Matrix
a.If A^2=0 show that A is not Invertible
b. If A^2= A and A is not the identity, show that A is not Invertible.

plz help.. i couldnt seem to figure it out?
a) (proof by contradiction) Suppose A were invertible. Then, multiplying both sides by $\displaystyle A^{-1}$, $\displaystyle A^{-1}A^2= A= 0$ which is a contradiction because 0 is not invertible.

Do exactly the same thing for b.