Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
Let $X$ and $Y$ be two matrices different from $I$, such that $XY = YX$ and $X^n - Y^n$ is invertible for some natural number $n$. If $X^n - Y^n = X^{n+1} - Y^{n+1} = X^{n+2} - Y^{n+2}$, then:
I - X is singular
I - Y is singular
X + Y = XY + I
(I - X)(I - Y) is non singular
Step-by-Step Solution
Key Concept: The invertibility of $X^n - Y^n$ combined with the equality of consecutive differences forces the commuting matrices $X$ and $Y$ to satisfy $X + Y = I + XY$, making both $I-X$ and $I-Y$ singular.
Given $X^n = Y^n + (X^n - Y^n)$ and the equality $X^n - Y^n = X^{n+1} - Y^{n+1} = X^{n+2} - Y^{n+2}$, we have $X^{n+1} - Y^{n+1} = X^n - Y^n$, which gives $X·X^n - Y·Y^n = X^n - Y^n$. This means $X^n(X-I) = Y^n(Y-I)$. Since $XY = YX$, matrices $X$ and $Y$ commute and share eigenvectors. The condition $X^n - Y^n = X^{n+1} - Y^{n+1}$ implies $(X^n - Y^n)(I - X + Y) = 0$. Since $X^n - Y^n$ is invertible, we get $I - X + Y = 0$, so $X + Y = I + XY$. This forces both $I-X$ and $I-Y$ to be singular (as $X$ and $Y$ must have eigenvalue 1 in their spectra).
Correct Answer: 1,2,3