Let A be a real 2× 2 matrix that is not a scalar multiple of I and satisfies A^2-3A+2I=O. Then det (A^6-3A^-1) equals

The minimal polynomial of $A$ divides $x^2-3x+2=(x-1)(x-2)$. Since $A$ is not scalar, its minimal polynomial cannot be linear, so it equals $(x-1)(x-2)$ and $A$ is diagonalisable with eigenvalues exactly $1$ and $2$. The matrix $A^6-3A^{-1}$ is a polynomial in $A$ and $A^{-1}$, so its eigenvalues are obtained by substituting the eigenvalues of $A$. For $\lambda=1$: $1-3=-2$. For $\lambda=2$: $64-\frac32=\frac{125}{2}$. The determinant is the product, $(-2)\cdot\frac{125}{2}=-125$. The option $-122$ comes from wrongly using $3I$ in place of $3A^{-1}$, which gives $(1-3)(64-3)$; $125$ drops the sign; and $-130$ is a distractor from arithmetic slips.