Let A be an n x n matrix and let I be the n x n identity matrix. If A3 = 0, show
ID: 3035045 • Letter: L
Question
Let A be an n x n matrix and let I be the n x n identity matrix. If A3 = 0, show that I – A is invertible and that (I – A) -1 = I + A + A2 . Use the above result to find the inverse of the following matrix. [Take the given matrix to be I – A. Show that A3 = 0, and use the result given above to find its inverse. No credit for calculating the inverse by any other method.]
1. Let A be an n x n matrix and let Ibe the n x n identity matrix. If A3 0, show that I-A is invertible and that (I-A) 1 I+ A A2. 1 2 -1 Use the above result to find the inverse of o 1 3 0 0 1 Take the given matrix to be I-A. Show that A 0, and use the result given above to find its inverse. No credit for calculating the inverse by any other method.]Explanation / Answer
We have (I-A)(I+A +A2) = I+A+A2–A-A2–A3 = I -0 = I . Also, (I+A +A2) (I-A) = I-A +A –A2 +A2-A3 = I-0 = I. Therefore, by the definition of inverse, (I – A)-1 = I + A + A2
Let I-A =
1
2
-1
0
1
3
0
0
1
Then A =
0
-2
1
0
0
-3
0
0
0
So that A2 = A.A =
0
0
6
0
0
0
0
0
0
And A3 = A2 . A=
0
0
0
0
0
0
0
0
0
Then (I-A)-1 , i.e. the inverse of the given matrix = I +A +A2 =
1
-2
7
0
1
-3
0
0
1
1
2
-1
0
1
3
0
0
1
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