Question

A square matrix A is said to be idempotent iff A2 = A. (i) Show that if A is idempotent, then so is I − A. (ii) Show that if A is idempotent, then the matrix 2A − I is also invertible. Hint: Same as before, guess the inverse and check your answer with the definition of inverse.

Answer 1

Explanation:

Given that A is a square matrix and A is idempotent

`A^2 = A`

Consider I-A

i)
`(I-A)^2 = (I-A).(I-A)\\= I^2 -2A.I+A^2\\= I-2A+A\\=I-A`

It follows that I-A is also idempotent

ii) Consider the matrix 2A-I

`(2A-I).(2A-I)=\\4A^2-4AI+I^2\\= 4A-4A+I\\=I`

So it follows that 2A-I matrix is its own inverse.