If A= bmatrix 1&0 1&1 bmatrix and I the identity, then for all n 1 (a) Aⁿ=nA-(n-1)I (b) Aⁿ=2ⁿ⁻¹A-(n-1)I (c) Aⁿ=nA+(n-1)I (d) Aⁿ=2ⁿ⁻¹A+(n-1)I

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[JEE Main 2005] If $A=\begin{bmatrix}1&0\\1&1\end{bmatrix}$ and $I$ the identity, then for all $n\ge1$

VAVidaara Admin Asked 1mo ago 1 views 1 answer

If $A=\begin{bmatrix}1&0\\1&1\end{bmatrix}$ and $I$ the identity, then for all $n\ge1$

(a) $A^n=nA-(n-1)I$
(b) $A^n=2^{n-1}A-(n-1)I$
(c) $A^n=nA+(n-1)I$
(d) $A^n=2^{n-1}A+(n-1)I$

VAVidaara Admin ✓ Vidaara Team ✓ Accepted · 1mo ago ▲ 0

Correct answer: (a) $A^n=nA-(n-1)I$

$A^n=\begin{bmatrix}1&0\\n&1\end{bmatrix}=nA-(n-1)I$, verified by induction.

JEE Main 2005 · Binomial Theorem — verified solution by the Vidaara Team.

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