If matrix $X = \begin{bmatrix} a & 1 \\ -a^2+a-1 & 1-a \end{bmatrix}$ and $X^2 - X + I = O$ ($I$ is the identity matrix and $O$ is the zero matrix), then the inverse of $X$ is
Given, $X^2 - X + I = O$
$\quad \implies X^2 = X - I$
$\quad \implies (X^{-1}) (X^2 ) = (X^{-1})(X - I )$ (Multiplying $X^{-1}$ on both sides)
$\quad \implies X = I - X^{-1} $
$\quad \implies X^{-1} = I - X$
Option (B)
@Satbir
$X^{-1} = I - X$
When i subtract the matrix $'X'$ from the identity matrix, then i got the inverse
$X^{-1} = \begin{bmatrix} 1-a &1 \\ -a^{2} + a -1 & a \end{bmatrix}$
which is not equal to the option $(B).$
Please correct me if i'm wrong?
Ohh sorry, was in the night I just subtract the diagonal elements.
Now I got it, it was just a silly mistake.
Thank you so mcuh@Satbir
It's a very simple question, We need to calculate the inverse of a 2x2 matrix, Inverse of a matrix $A = A^{-1} = \frac{Adjoint(A)}{|A|}$ Adjoint(A) = [cofactors of A]T, but for 2x2 matrix we have direct formula: A = a b c d is a 2x2 matrix then Adjoint of A = d -b -c a |A| = ad-bc So answer is (B)!
You may not evaluate the given equation $X^{2} - X + I = O$.
Rather start evaluating from the choices given. $X.X^{-1} = I$ and the first element should be 1.
You may verify the full multiplication for option B to see whether it indeed yields the identity matrix.
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