Question

1 A= Find A1017 and Justify your answer 1 ✓2 B = O 01 3 Find B” and Justify your answer. 0 3

Answer 1

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1 A= - - R] 0 A = 2 A4 = A2A2, 0 -1 0+ 01 [O- 10 +0 -1 + 이 A1 = A1016 A = (A4)254 A = (-1)254 1354 A = (1)I2 A = A = 1] = (-1) 1 1 = (-1) 1017

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$\small B=\begin{bmatrix} \frac{1}{3} & \frac{1}{3}\\ 0 & \frac{1}{3} \end{bmatrix}=\frac{1}{3}\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}\\ B^2=\begin{bmatrix} \frac{1}{3} & \frac{1}{3}\\ 0 & \frac{1}{3} \end{bmatrix}\begin{bmatrix} \frac{1}{3} & \frac{1}{3}\\ 0 & \frac{1}{3} \end{bmatrix}=\begin{bmatrix} \frac{1}{3^2} & \frac{2}{3^2}\\ 0 & \frac{1}{3^2} \end{bmatrix}=\frac{1}{3^2}\begin{bmatrix} 1 & 2\\ 0 & 1 \end{bmatrix}\\ B^3=\frac{1}{3}\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}\frac{1}{3^2}\begin{bmatrix} 1 & 2\\ 0 & 1 \end{bmatrix}=\frac{1}{3^3}\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 2\\ 0 & 1 \end{bmatrix}=\frac{1}{3^3}\begin{bmatrix} 1 & 3\\ 0 & 1 \end{bmatrix}\\ $So we claim that $\\ B^n=\frac{1}{3^n}\begin{bmatrix} 1 & n\\ 0 & 1 \end{bmatrix}\\ $To prove that we use principal of induction\\ Let it be true for n=k$\\ B^k=\frac{1}{3^k}\begin{bmatrix} 1 & k\\ 0 & 1 \end{bmatrix}\\$

$\small B^{k+1}=B^kB=B^n=\frac{1}{3^k}\begin{bmatrix} 1 & k\\ 0 & 1 \end{bmatrix}\frac{1}{3}\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}=\frac{1}{3^{k+1}}\begin{bmatrix} 1 & k+1\\ 0 & 1 \end{bmatrix}\\ $Then the statement is true for n=k+1 if it be true for n=k\\ Therefore by principal of induction$\\ B^n=\frac{1}{3^n}\begin{bmatrix} 1 & n\\ 0 & 1 \end{bmatrix}$

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