Question

If A = $\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$ or $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then which of the following holds of all n ³ 1,

• A. Aⁿ = nA – (n – 1) I
• B. Aⁿ = 2^n–1 A – (n – 1)I
• C. Aⁿ = nA + (n – 1)I
• D. Aⁿ = 2^n–1 A + (n – 1) I

Answer 1

Answer: (A)

Aⁿ = nA – (n – 1) I

Answer 2

=$\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$̃ Aⁿ = nA – (n – 1) I

̃ A = nA – (n – 1) A = A

A² = $\begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$

A^m+1 = A^m. A = (mA – (m– 1) A) A

= mA² – mA + A

= m $\begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$– m A + $\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$

= $\begin{bmatrix} m + 1 & 0 \\ m + 1 & m + 1 \end{bmatrix}$ – m $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

= (m + 1)A – mA = A

So (1) is true