高専の数学解説

第17章 17.12

(1)

A=(2−1−1−141−112)A = \begin{pmatrix} 2 & -1 & -1 \\ -1 & 4 & 1 \\ -1 & 1 & 2 \end{pmatrix} とする と

∣A−λE∣=∣2−λ−1−1−14−λ1−112−λ∣=(2−λ)2(4−λ)+1+1−(4−λ)−2(2−λ)|A - \lambda E| = \begin{vmatrix} 2-\lambda & -1 & -1 \\ -1 & 4-\lambda & 1 \\ -1 & 1 & 2-\lambda \end{vmatrix} = (2-\lambda)^2(4-\lambda) + 1 + 1 - (4-\lambda) - 2(2-\lambda)

=(4−λ)(4−4λ+λ2−1)+2−4+2λ=(4−λ)(3−4λ+λ2)−2(1−λ)=(4−λ)(1−λ)(3−λ)−2(1−λ)= (4-\lambda)(4-4\lambda + \lambda^2 - 1) + 2 - 4 + 2\lambda = (4-\lambda)(3 - 4\lambda + \lambda^2) - 2(1-\lambda) = (4-\lambda)(1-\lambda)(3-\lambda) - 2(1-\lambda)

=(1−λ)(12−7λ+λ2−2)=(1−λ)(10−7λ+λ2)=(1−λ)(2−λ)(5−λ)λ=1,2,5= (1-\lambda)(12 - 7\lambda + \lambda^2 - 2) = (1-\lambda)(10 - 7\lambda + \lambda^2) = (1-\lambda)(2-\lambda)(5-\lambda) \quad \lambda = 1,2,5

λ=1\lambda = 1 のとき

(1−1−1−131−111)→(1−1−1020000)→(1−1−1010000)→(10−1010000)p1=C11(101)\begin{pmatrix} 1 & -1 & -1 \\ -1 & 3 & 1 \\ -1 & 1 & 1 \end{pmatrix} \to \begin{pmatrix} 1 & -1 & -1 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & -1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad p_1 = C_1^1 \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}

λ=2\lambda = 2 のとき

(0−1−1−121−110)→(1−10−1210−1−1)→(1−10011011)→(1−10011000)→(101011000)p2=C21(−1−11)\begin{pmatrix} 0 & -1 & -1 \\ -1 & 2 & 1 \\ -1 & 1 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & -1 & 0 \\ -1 & 2 & 1 \\ 0 & -1 & -1 \end{pmatrix} \to \begin{pmatrix} 1 & -1 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 1 \end{pmatrix} \to \begin{pmatrix} 1 & -1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix} \quad p_2 = C_2^1 \begin{pmatrix} -1 \\ -1 \\ 1 \end{pmatrix}

λ=5\lambda = 5 のとき

(−3−1−1−1−11−11−3)→(11−1−3−1−1−11−3)→(11−102−402−4)→(11−101−2000)→(10101−2000)p3=C31(−121)\begin{pmatrix} -3 & -1 & -1 \\ -1 & -1 & 1 \\ -1 & 1 & -3 \end{pmatrix} \to \begin{pmatrix} 1 & 1 & -1 \\ -3 & -1 & -1 \\ -1 & 1 & -3 \end{pmatrix} \to \begin{pmatrix} 1 & 1 & -1 \\ 0 & 2 & -4 \\ 0 & 2 & -4 \end{pmatrix} \to \begin{pmatrix} 1 & 1 & -1 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \end{pmatrix} \quad p_3 = C_3^1 \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}

(2)

A=(10−1222212)A = \begin{pmatrix} 1 & 0 & -1 \\ 2 & 2 & 2 \\ 2 & 1 & 2 \end{pmatrix} とする と

∣A−λE∣=∣1−λ0−122−λ2212−λ∣=(1−λ)(2−λ)2−2+2(2−λ)−2(1−λ)|A - \lambda E| = \begin{vmatrix} 1-\lambda & 0 & -1 \\ 2 & 2-\lambda & 2 \\ 2 & 1 & 2-\lambda \end{vmatrix} = (1-\lambda)(2-\lambda)^2 - 2 + 2(2-\lambda) - 2(1-\lambda)

=(1−λ)(4−4λ+λ2−2)−2+4−2λ=(1−λ)(2−4λ+λ2)+2(1−λ)=(1−λ)(4−4λ+λ2)= (1-\lambda)(4 - 4\lambda + \lambda^2 - 2) - 2 + 4 - 2\lambda = (1-\lambda)(2 - 4\lambda + \lambda^2) + 2(1-\lambda) = (1-\lambda)(4 - 4\lambda + \lambda^2)

=(1−λ)(2−λ)2λ=1,2(2は重解)= (1-\lambda)(2-\lambda)^2 \quad \lambda = 1, 2 \quad (2 \text{は重解})

λ=1\lambda = 1 のとき

(00−1212211)→(211212001)→(211001001)→(210001000)→(11/20001000)p1=C1(−1/210)\begin{pmatrix} 0 & 0 & -1 \\ 2 & 1 & 2 \\ 2 & 1 & 1 \end{pmatrix} \to \begin{pmatrix} 2 & 1 & 1 \\ 2 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix} \to \begin{pmatrix} 2 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix} \to \begin{pmatrix} 2 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & 1/2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \quad p_1 = C_1 \begin{pmatrix} -1/2 \\ 1 \\ 0 \end{pmatrix}

λ=2\lambda = 2 のとき

(−10−1202210)→(101101210)→(10100001−2)→(10101−2000)p2=C2(−121)\begin{pmatrix} -1 & 0 & -1 \\ 2 & 0 & 2 \\ 2 & 1 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & 0 & 1 \\ 1 & 0 & 1 \\ 2 & 1 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 1 & -2 \end{pmatrix} \to \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \end{pmatrix} \quad p_2 = C_2 \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}

(3)

A=(21−1110−101)とするときA = \begin{pmatrix} 2 & 1 & -1 \\ 1 & 1 & 0 \\ -1 & 0 & 1 \end{pmatrix} \text{とするとき}

∣A−λE∣=∣2−λ1−111−λ0−101−λ∣=(2−λ)(1−λ)2−2(1−λ)=(1−λ)(2−3λ+λ2−2)|A - \lambda E| = \begin{vmatrix} 2-\lambda & 1 & -1 \\ 1 & 1-\lambda & 0 \\ -1 & 0 & 1-\lambda \end{vmatrix} = (2-\lambda)(1-\lambda)^2 - 2(1-\lambda) = (1-\lambda)(2-3\lambda + \lambda^2 - 2)

=(1−λ)(λ2−3λ)=λ(1−λ)(λ−3)λ=0,1,3= (1-\lambda)(\lambda^2 - 3\lambda) = \lambda(1-\lambda)(\lambda - 3) \quad \lambda = 0, 1, 3

λ=0\lambda = 0 のとき

(21−1110−101)→(11021−1−101)→(1100−1−1011)→(110011000)→(10−1011000)p1=C1(1−11)\begin{pmatrix} 2 & 1 & -1 \\ 1 & 1 & 0 \\ -1 & 0 & 1 \end{pmatrix} \to \begin{pmatrix} 1 & 1 & 0 \\ 2 & 1 & -1 \\ -1 & 0 & 1 \end{pmatrix} \to \begin{pmatrix} 1 & 1 & 0 \\ 0 & -1 & -1 \\ 0 & 1 & 1 \end{pmatrix} \to \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & 0 & -1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix} \quad p_1 = C_1 \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}

λ=1\lambda = 1 のとき

(11−1100−100)→(11−1100000)→(10011−1000)→(10001−1000)p2=C3(011)\begin{pmatrix} 1 & 1 & -1 \\ 1 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & 1 & -1 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & -1 \\ 0 & 0 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \end{pmatrix} \quad p_2 = C_3 \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}

λ=3\lambda = 3 のとき

(−11−11−20−10−2)→(1−111−20102)→(1−110−1−1011)→(1−11011000)→(102011000)p3=C3(−2−11)\begin{pmatrix} -1 & 1 & -1 \\ 1 & -2 & 0 \\ -1 & 0 & -2 \end{pmatrix} \to \begin{pmatrix} 1 & -1 & 1 \\ 1 & -2 & 0 \\ 1 & 0 & 2 \end{pmatrix} \to \begin{pmatrix} 1 & -1 & 1 \\ 0 & -1 & -1 \\ 0 & 1 & 1 \end{pmatrix} \to \begin{pmatrix} 1 & -1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix} \to \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix} \quad p_3 = C_3 \begin{pmatrix} -2 \\ -1 \\ 1 \end{pmatrix}

(4)

A=(32−223−2−2−23)とするとA = \begin{pmatrix} 3 & 2 & -2 \\ 2 & 3 & -2 \\ -2 & -2 & 3 \end{pmatrix} \text{とすると}

∣A−λE∣=∣3−λ2−223−λ−2−2−23−λ∣=(3−λ)3+8+8−12(3−λ)=27−27λ+9λ2−λ3+16−36+12λ|A - \lambda E| = \begin{vmatrix} 3-\lambda & 2 & -2 \\ 2 & 3-\lambda & -2 \\ -2 & -2 & 3-\lambda \end{vmatrix} = (3-\lambda)^3 + 8 + 8 - 12(3-\lambda) = 27 - 27\lambda + 9\lambda^2 - \lambda^3 + 16 - 36 + 12\lambda

=−λ3+9λ2−15λ+7=−(λ−1)(λ−1)(λ−7)λ=1,7 （１は重解）= -\lambda^3 + 9\lambda^2 - 15\lambda + 7 = -(\lambda - 1)(\lambda - 1)(\lambda - 7) \quad \lambda = 1, 7 \text{ （１\text{は重解}）}

λ=1\lambda = 1 のとき

(22−222−2−2−22)→(11−1000000)p1=C1(−110)+C2(101)\begin{pmatrix} 2 & 2 & -2 \\ 2 & 2 & -2 \\ -2 & -2 & 2 \end{pmatrix} \to \begin{pmatrix} 1 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \quad p_1 = C_1 \begin{pmatrix} -1 \\ 1 \\ 0 \end{pmatrix} + C_2 \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}

λ=7\lambda = 7 のとき

(−42−22−4−2−2−2−4)→(−21−11−2−1112)→(1121−2−1−21−1)→(1120−3−3033)→(112011000)\begin{pmatrix} -4 & 2 & -2 \\ 2 & -4 & -2 \\ -2 & -2 & -4 \end{pmatrix} \to \begin{pmatrix} -2 & 1 & -1 \\ 1 & -2 & -1 \\ 1 & 1 & 2 \end{pmatrix} \to \begin{pmatrix} 1 & 1 & 2 \\ 1 & -2 & -1 \\ -2 & 1 & -1 \end{pmatrix} \to \begin{pmatrix} 1 & 1 & 2 \\ 0 & -3 & -3 \\ 0 & 3 & 3 \end{pmatrix} \to \begin{pmatrix} 1 & 1 & 2 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix}

→(101011000)p2=C3(−1−11)\to \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{pmatrix} \quad p_2 = C_3 \begin{pmatrix} -1 \\ -1 \\ 1 \end{pmatrix}

解説: r31bn1z