If A=1 23 4, then 8A−4=

The characteristic equation is 1−λ234−λ=0⇒λ2−5λ−2=0∴A2−5A−2I=0⇒I−5A−1−2A−2=0⇒A−2=12I−5A−1⇒A−4=14I−5A−12=14I−10A−1+25A−2=14I−10A−1+252I−5A−1∴A−4=1827I−145A−1

If A=1 23 4, then 8A−4=

The characteristic equation is 1−λ234−λ=0⇒λ2−5λ−2=0∴A2−5A−2I=0⇒I−5A−1−2A−2=0⇒A−2=12I−5A−1⇒A−4=14I−5A−12=14I−10A−1+25A−2=14I−10A−1+252I−5A−1∴A−4=1827I−145A−1

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If $A = [\begin{array}{ll} 1 2 \\ 3 4 \end{array}],$ then $8 A^{- 4} =$

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$27 I - 145 A^{- 1}$

$145 A^{- 1} + 27 I$

$145 A^{- 1} - 27 I$

$29 A^{- 1} + 9 I$

answer is C.

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Detailed Solution

The characteristic equation is $|\begin{array}{cc} 1 - λ & 2 \\ 3 & 4 - λ \end{array}| = 0$

$\begin{array}{l} \Rightarrow λ^{2} - 5 λ - 2 = 0 \\ ∴ A^{2} - 5 A - 2 I = 0 \Rightarrow I - 5 A^{- 1} - 2 A^{- 2} = 0 \\ \Rightarrow A^{- 2} = \frac{1}{2} [I - 5 A^{- 1}] \Rightarrow A^{- 4} = \frac{1}{4} {[I - 5 A^{- 1}]}^{2} \\ = \frac{1}{4} [I - 10 A^{- 1} + 25 A^{- 2}] \\ = \frac{1}{4} [I - 10 A^{- 1} + \frac{25}{2} (I - 5 A^{- 1})] \\ ∴ A^{- 4} = \frac{1}{8} [27 I - 145 A^{- 1}] \end{array}$