For the matrices A=[array|c|c| 3 & -4 1 & -1 array] and B=[array|c|c| -29 & 49 -13 & 18 ar…

Question

For the matrices $A=[\begin{array}{|c|c|} 3 & -4 \ 1 & -1 \end{array}]$ and $B=[\begin{array}{|c|c|} -29 & 49 \ -13 & 18 \end{array}]$, if $({A}^{15}+B)[\begin{gathered} x \ y \end{gathered}]=[\begin{gathered} 0 \ 0 \end{gathered}]$, then among the following which one is true?

Accepted Answer

$\begin{gathered} A=[\begin{gathered} 3 \quad -4 \ 1 \quad -1],B=[23 \quad 49 \ 45 \quad 21] \ \quad (3-\lambda )(-1-\lambda )+4=0 \ (\lambda -3)(\lambda +1)+4={\lambda }^{2}-2\lambda +=0 \ \Rightarrow {A}^{2}-2A+I=0 \ {A}^{2}=2A-I,{A}^{3}=2{A}^{2}-A=2(2A-I)-A=3A-2I \ {A}^{4}=4{A}^{2}+I-4A=4(2A-I)-4A+I \ =4(2A-I)-3A=5A-4I \ {({A}^{5})}^{3}=(5A-4I{)}^{3} \ =125{A}^{3}-3	imes 25{A}^{2}(4)+3(5A)({4}^{2})-64I \ =125(3A-2I)-300(2A-I)+240A-64I \ =A(375-600+240)+I(-250+300-64) \ =15A-14I \ \Rightarrow {A}^{15}+B=15A-14I+B \ =[2 \quad -11 \ 1 \quad -11]\, ({A}^{15}+B)[x \ y]=[0 \ 0]\Rightarrow 2x=11y \end{gathered} \end{gathered}$

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