A square matrix is idempotent if AA=A. Show that if A is idempotent, then the matrix 2A-I is invertible and equal to its own inverse. Solution A square matrix is idempotent if AA=A. Show that if A is idempotent, then the matrix 2A-I is invertible and equal to its own inverse. consider (2A-I)(2A-I) = 4A*A - 2A - 2A + I = 4A - 2A -2A + I = 4A-4A + I = I so (2A-I) is nothing but inverse and it is equal to its own matrix..

1. A square matrix is idempotent if AA=A. Show that if A is idempotent, then the matrix 2A-I is
invertible and equal to its own inverse.
Solution
A square matrix is idempotent if AA=A. Show that if A is idempotent, then the matrix 2A-I is
invertible and equal to its own inverse.
consider (2A-I)(2A-I)
= 4A*A - 2A - 2A + I
= 4A - 2A -2A + I
= 4A-4A + I
= I
so (2A-I) is nothing but inverse and it is equal to its own matrix.