IF A=1 01 1 and I=1 00 1, which one of the following holds for all n≥1,, (by the principal of mathematical induction)

A2=10111011=1021A3=10211011=1031∴ An=1 0n 1nA=n 0n n,(n−1)I=n−100n−1nA−(n−1)I=1 0n 1=An

IF A=1 01 1 and I=1 00 1, which one of the following holds for all n≥1,, (by the principal of mathematical induction)

A2=10111011=1021A3=10211011=1031∴ An=1 0n 1nA=n 0n n,(n−1)I=n−100n−1nA−(n−1)I=1 0n 1=An

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IF $A = [\begin{array}{ll} 1 0 \\ 1 1 \end{array}] and I = [\begin{array}{ll} 1 0 \\ 0 1 \end{array}]$ , which one of the following holds for all $n \geq 1$ ,, (by the principal of mathematical induction)

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

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

$A^{n} = 2^{n - 1} A - (n - 1) I$

$A^{n} = 2^{n - 1} A + (n - 1) I$

answer is C.

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Detailed Solution

$\begin{aligned} A^{2} = [\begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array}] [\begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array}] = [\begin{array}{ll} 1 & 0 \\ 2 & 1 \end{array}] \\ A^{3} = [\begin{array}{ll} 1 & 0 \\ 2 & 1 \end{array}] [\begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array}] = [\begin{array}{ll} 1 & 0 \\ 3 & 1 \end{array}] \end{aligned}$