Question

If A is idempotent show that ( I - 2A) is equal to its inverse

Answer 1

Well a matrix A is idempotent if AA=A, and to show an inverse, we need to multiply some things together. Consider \((I-2A)(I-2A)\). If we FOIL that expression, we get\[I^2-4A+4A^2\]But since \(A\) is idempotent, \(A^2=A\). So we can simplify the expression as\[I^2-4A+4A^2=I-4A+4A=I.\]Thus, \((I-2A)^{-1}=(I-2A)\).