Let be an matrix such that

MATHEMATICS · Subjective · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

Let

A

be an

n\times n

matrix such that

A^{2}=2A

. (i) Show that

A(I_n-A)=I_nA-A^2

. (ii) Using the result of part (i), prove that the matrix

I_n-A

is invertible. (iii) Determine

(I_n-A)^{-1}

in its simplest form.