If $A=\begin{bmatrix}1 & 0 \\ 1 & 1\end{bmatrix}$ and $I=\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$, then which one of the following holds for all $n \geq 1$, by the principle of mathematical induction

JEE Main · 2005 · 08 May 2005 · Offline · Mathematical Induction

A = [1101]

and

I = [1001]

, then which one of the following holds for all

n \geq 1

, by the principle of mathematical induction

Answer: (A)

Check the base case:

A^{1} = A = 1 \cdot A - 0 \cdot I

. Now assume

A^{k} = k A - (k - 1) I

. A direct computation gives

A^{2} = (1201) = 2 A - I

, so

A^{k + 1} = A \cdot A^{k} = k A^{2} - (k - 1) A = k (2 A - I) - (k - 1) A = (k + 1) A - k I

. By induction,

A^{n} = n A - (n - 1) I

for every

n \geq 1

JEE Main · 2005 · 08 May 2005 · Offline · Single correct option