If matrix X = [[a, a-1], [1-a, 1-a]] and X^2 - X + I = O (I is the identity matrix and O is the zero matrix), then the i...

Question

If matrix X = [[a, a-1], [1-a, 1-a]] and X^2 - X + I = O (I is the identity matrix and O is the zero matrix), then the inverse of X is
A. [[1-a, a-1], [a, 1-a]]
B. [[1-a, a-1], [a-1, a]]
C. I - X
D. X has full rank

Accepted Answer

X^(-1) = I - X = [[1-a, 1-a],[a-1, a]], which corresponds to option B. We are given a 2x2 matrix X satisfying the polynomial equation X^2 - X + I = O. From this relation we can algebraically derive the inverse of X. If X^2 - X + I = O, then multiplying both sides by X^(-1) gives X - I + X^(-1) = O, so X^(-1) = I - X.

If matrix X = [[a, a-1], [1-a, 1-a]] and X^2 - X + I = O (I is the identity matrix and O is the zero matrix), then the inverse of X is A. [[1-a, a-1], [a, 1-a]] B. [[1-a, a-1], [a-1, a]] C. I - X D. X has full rank