Prove that if A is a squre matrix of order 'n' and A³=0, this matrix is invertible: (½)A²+A+I_n?
Prove that this matrix is invertible:
(½)A²+A+I_n
if A³=0, in which I_n is identity matrix of order 'n'.
A is a square matrix of n×n.
More Questions From This User See All
Answers & Comments
Verified answer
The inverse is (1/2)A^2-A+I_n.
In order to find it, suppose the inverse is a polynomial in A. Then this polynomial is at most of degree 2, since A^3=0 means that every power greater than 2 is zero. Write C:=a*A^2+b*A+c*I_n and find a, b, c such that BC=I_n where B=(1/2)A^2+A+I_n is our matrix (note that, since C is polynomial in A, it is automatically BC=CB).
This proposition is wrong, Here's a counterexample:
Let n=2
[A] be the 2 X 2 zero matrix, then [A][A][A]=0, and [A] is not invertible.
Multiply
[I+A+(1/2)A^2]
and
[I-A+(1/2)A^2)]
and see what you get.