Question:medium
If $A$ is a non-singular matrix of order $n$ satisfying the matrix equation $I+A+A^{2}+A^{3}+...+A^{10}=0$, where $I$ and $0$ are, respectively, unit and null matrices of order $n$, then $A^{10}=$
If $A$ is a non-singular matrix of order $n$ satisfying the matrix equation $I+A+A^{2}+A^{3}+...+A^{10}=0$, where $I$ and $0$ are, respectively, unit and null matrices of order $n$, then $A^{10}=$
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Logic Tip: A geometric series of matrices $I + A + A^2 + \dots + A^k = 0$ behaves identically to a geometric series of numbers. Multiplying by $(I - A)$ yields $I - A^{k+1} = 0$, meaning $A^{11} = I$. Multiplying by $A^{-1}$ immediately gives $A^{10} = A^{-1}$.
Updated On: Apr 27, 2026
- $A^{-1}$
- $I$
- $A$
- $I+A$
- $0$
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The Correct Option is A
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