高専の数学解説

第16章 16.15

(1)

∣111xx2x3x3x4x5∣=∣111x−1x2−1x3−10x3−1x4−1∣=∣x−1x2−1x3−1x4−1∣\begin{vmatrix} 1 & 1 & 1 \\ x & x^2 & x^3 \\ x^3 & x^4 & x^5 \end{vmatrix} = \begin{vmatrix} 1 & 1 & 1 \\ x-1 & x^2-1 & x^3-1 \\ 0 & x^3-1 & x^4-1 \end{vmatrix} = \begin{vmatrix} x-1 & x^2-1 \\ x^3-1 & x^4-1 \end{vmatrix}

=(x−1)2∣1x+1x2+x+1x3+x2+x+1∣=(x−1)2{x3+x2+x+1−(x+1)(x2+x+1)}= (x-1)^2 \begin{vmatrix} 1 & x+1 \\ x^2+x+1 & x^3+x^2+x+1 \end{vmatrix} = (x-1)^2 \{x^3+x^2+x+1-(x+1)(x^2+x+1)\}

=(x−1)2(x3+x2+x+1−x3−2x2−2x−1)=(x−1)2(−x2−x)= (x-1)^2(x^3+x^2+x+1-x^3-2x^2-2x-1) = (x-1)^2(-x^2-x)

=−x(x+1)(x−1)2= -x(x+1)(x-1)^2

(2)

∣x−2−111x−411−1x−2∣=(x−2)(x−4)(x−2)−1−1−(x−4)+(x−2)+(x−2)\begin{vmatrix} x-2 & -1 & 1 \\ 1 & x-4 & 1 \\ 1 & -1 & x-2 \end{vmatrix} = (x-2)(x-4)(x-2)-1-1-(x-4)+(x-2)+(x-2)

=x3−8x2+20x−16+x−2=x3−8x2+21x−18= x^3-8x^2+20x-16+x-2 = x^3-8x^2+21x-18

=(x−2)(x2−6x+9)=(x−2)(x−3)2= (x-2)(x^2-6x+9) = (x-2)(x-3)^2

(3)

∣1cos⁡αcos⁡2α1cos⁡βcos⁡2β1cos⁡γcos⁡2γ∣=∣1cos⁡αcos⁡2α0cos⁡β−cos⁡αcos⁡2β−cos⁡2α0cos⁡γ−cos⁡αcos⁡2γ−cos⁡2α∣\begin{vmatrix} 1 & \cos\alpha & \cos 2\alpha \\ 1 & \cos\beta & \cos 2\beta \\ 1 & \cos\gamma & \cos 2\gamma \end{vmatrix} = \begin{vmatrix} 1 & \cos\alpha & \cos 2\alpha \\ 0 & \cos\beta-\cos\alpha & \cos 2\beta-\cos 2\alpha \\ 0 & \cos\gamma-\cos\alpha & \cos 2\gamma-\cos 2\alpha \end{vmatrix}

=∣cos⁡β−cos⁡α2(cos⁡2β−cos⁡2α)cos⁡γ−cos⁡α2(cos⁡2γ−cos⁡2α)∣=2∣cos⁡β−cos⁡α(cos⁡β−cos⁡α)(cos⁡β+cos⁡α)cos⁡γ−cos⁡α(cos⁡γ−cos⁡α)(cos⁡γ+cos⁡α)∣= \begin{vmatrix} \cos\beta-\cos\alpha & 2(\cos^2\beta-\cos^2\alpha) \\ \cos\gamma-\cos\alpha & 2(\cos^2\gamma-\cos^2\alpha) \end{vmatrix} = 2 \begin{vmatrix} \cos\beta-\cos\alpha & (\cos\beta-\cos\alpha)(\cos\beta+\cos\alpha) \\ \cos\gamma-\cos\alpha & (\cos\gamma-\cos\alpha)(\cos\gamma+\cos\alpha) \end{vmatrix}

=2(cos⁡β−cos⁡α)(cos⁡γ−cos⁡α)∣1cos⁡β+cos⁡α1cos⁡γ+cos⁡α∣= 2(\cos\beta-\cos\alpha)(\cos\gamma-\cos\alpha) \begin{vmatrix} 1 & \cos\beta+\cos\alpha \\ 1 & \cos\gamma+\cos\alpha \end{vmatrix}

=2(cos⁡β−cos⁡α)(cos⁡γ−cos⁡α)(cos⁡γ−cos⁡β)= 2(\cos\beta-\cos\alpha)(\cos\gamma-\cos\alpha)(\cos\gamma-\cos\beta)

(4)

∣x01x1xx00xx1x10x∣=∣x01x1xx00xx101−10∣\begin{vmatrix} x & 0 & 1 & x \\ 1 & x & x & 0 \\ 0 & x & x & 1 \\ x & 1 & 0 & x \end{vmatrix} = \begin{vmatrix} x & 0 & 1 & x \\ 1 & x & x & 0 \\ 0 & x & x & 1 \\ 0 & 1 & -1 & 0 \end{vmatrix}

=x∣xx0xx11−10∣−x∣xx1xx11−10∣=x(x+x)−(−x2+1−x2)= x \begin{vmatrix} x & x & 0 \\ x & x & 1 \\ 1 & -1 & 0 \end{vmatrix} - x \begin{vmatrix} x & x & 1 \\ x & x & 1 \\ 1 & -1 & 0 \end{vmatrix} = x(x+x) - (-x^2+1-x^2)

=2x2+2x2−1=4x2−1=(2x−1)(2x+1)= 2x^2 + 2x^2 - 1 = 4x^2 - 1 = (2x-1)(2x+1)

(5)

∣xabcaxbcabxcabcx∣=∣a+b+c+xabca+b+c+xxbca+b+c+xbxca+b+c+xbcx∣=(a+b+c+x)∣1abc1xbc1bxc1bcx∣\begin{vmatrix} x & a & b & c \\ a & x & b & c \\ a & b & x & c \\ a & b & c & x \end{vmatrix} = \begin{vmatrix} a+b+c+x & a & b & c \\ a+b+c+x & x & b & c \\ a+b+c+x & b & x & c \\ a+b+c+x & b & c & x \end{vmatrix} = (a+b+c+x)\begin{vmatrix} 1 & a & b & c \\ 1 & x & b & c \\ 1 & b & x & c \\ 1 & b & c & x \end{vmatrix} =(a+b+c+x)∣1abc0x−a000b−ax−b00b−ac−bx−c∣=(a+b+c+x)∣x−a00b−ax−b0b−ac−bx−c∣= (a+b+c+x)\begin{vmatrix} 1 & a & b & c \\ 0 & x-a & 0 & 0 \\ 0 & b-a & x-b & 0 \\ 0 & b-a & c-b & x-c \end{vmatrix} = (a+b+c+x)\begin{vmatrix} x-a & 0 & 0 \\ b-a & x-b & 0 \\ b-a & c-b & x-c \end{vmatrix}

=(a+b+c+x)(x−a)(x−b)(x−c)= (a+b+c+x)(x-a)(x-b)(x-c)

解説: r31bn1z