# 基本的な行列の問題

## ■問題

$\left|\begin{array}{cccc}x& 1& 2& 3\\ 1& x+1& 4& 5\\ 2& 3& 2x& 6\\ 4& 5& 6& 2x+1\end{array}\right|$

## ■答

${4x}^{4}+6{x}^{3}-18{x}^{2}+150x-76$

## ■計算

$\left|\begin{array}{cccc}x& 1& 2& 3\\ 1& x+1& 4& 5\\ 2& 3& 2x& 6\\ 4& 5& 6& 2x+1\end{array}\right|$

$=1×{\left(-1\right)}^{1+2}\left|\begin{array}{ccc}1& 4& 5\\ 2& 2x& 6\\ 4& 6& 2x+1\end{array}\right|$ $+\left(x+1\right){\left(-1\right)}^{2+2}\left|\begin{array}{ccc}x& 2& 3\\ 2& 2x& 6\\ 4& 6& 2x+1\end{array}\right|$ $+3×{\left(-1\right)}^{3+2}\left|\begin{array}{ccc}x& 2& 3\\ 1& 4& 5\\ 4& 6& 2x+1\end{array}\right|$ $+5×{\left(-1\right)}^{4+2}\left|\begin{array}{ccc}x& 2& 3\\ 1& 4& 5\\ 2& 2x& 6\end{array}\right|$

$=-\left|\begin{array}{ccc}1& 4& 5\\ 2& 2x& 6\\ 4& 6& 2x+1\end{array}\right|$ $+\left(x+1\right)\left|\begin{array}{ccc}x& 2& 3\\ 2& 2x& 6\\ 4& 6& 2x+1\end{array}\right|$ $-3\left|\begin{array}{ccc}x& 2& 3\\ 1& 4& 5\\ 4& 6& 2x+1\end{array}\right|$ $+5\left|\begin{array}{ccc}x& 2& 3\\ 1& 4& 5\\ 2& 2x& 6\end{array}\right|$　 ･･･（1）

それぞれの項の行列式を計算する．

$\left|\begin{array}{ccc}1& 4& 5\\ 2& 2x& 6\\ 4& 6& 2x+1\end{array}\right|$

この計算においても行列式の値は第2列で展開(余因子展開)して求めることにする．

$=4×{\left(-1\right)}^{1+2}\left|\begin{array}{cc}2& 6\\ 4& 2x+1\end{array}\right|+2x×{\left(-1\right)}^{2+2}\left|\begin{array}{cc}1& 5\\ 4& 2x+1\end{array}\right|+6×{\left(-1\right)}^{3+2}\left|\begin{array}{cc}1& 5\\ 2& 6\end{array}\right|$

$=-4\left|\begin{array}{cc}2& 6\\ 4& 2x+1\end{array}\right|+2x\left|\begin{array}{cc}1& 5\\ 4& 2x+1\end{array}\right|-6\left|\begin{array}{cc}1& 5\\ 2& 6\end{array}\right|$

$=-4\left(4x+2-24\right)+2x\left(2x+1-20\right)-6\left(6-10\right)$

$=-8\left(2x-11\right)+2x\left(2x-19\right)-12\left(-2\right)$

$=-16x+88+4{x}^{2}-38x+24$

$=4{x}^{2}-54x+112$　 ･･･（2）

$\left|\begin{array}{ccc}x& 2& 3\\ 2& 2x& 6\\ 4& 6& 2x+1\end{array}\right|$

$=2×{\left(-1\right)}^{1+2}\left|\begin{array}{cc}2& 6\\ 4& 2x+1\end{array}\right|+2x×{\left(-1\right)}^{2+2}\left|\begin{array}{cc}x& 3\\ 4& 2x+1\end{array}\right|+6×{\left(-1\right)}^{3+2}\left|\begin{array}{cc}x& 3\\ 2& 6\end{array}\right|$

$=-2\left|\begin{array}{cc}2& 6\\ 4& 2x+1\end{array}\right|+2x\left|\begin{array}{cc}x& 3\\ 4& 2x+1\end{array}\right|-6\left|\begin{array}{cc}x& 3\\ 2& 6\end{array}\right|$

$=-2\left(4x+2-24\right)+2x\left({2x}^{2}+x-12\right)-6\left(6x-6\right)$

$=-2\left(4x-22\right)+2x\left({2x}^{2}+x-12\right)-6\left(6x-6\right)$

$=-8x+44+4{x}^{3}+2{x}^{2}-24x-36x+36$

$={4x}^{3}+{2x}^{2}-68x+80$　 ･･･（3）

$\left|\begin{array}{ccc}x& 2& 3\\ 1& 4& 5\\ 4& 6& 2x+1\end{array}\right|$

$=2×{\left(-1\right)}^{1+2}\left|\begin{array}{cc}1& 5\\ 4& 2x+1\end{array}\right|+4×{\left(-1\right)}^{2+2}\left|\begin{array}{cc}x& 3\\ 4& 2x+1\end{array}\right|+6×{\left(-1\right)}^{3+2}\left|\begin{array}{cc}x& 3\\ 1& 5\end{array}\right|$

$=-2|\begin{array}{cc}1& 5\\ 4& 2x+1\end{array}|+4|\begin{array}{cc}x& 3\\ 4& 2x+1\end{array}|-6|\begin{array}{cc}x& 3\\ 1& 5\end{array}|$

$=-2\left\{\left(2x+1-20\right)-2\left(2{x}^{2}+x-12\right)+3\left(5x-3\right)\right\}$

$=-2\left\{\left(2x-19\right)-2\left(2{x}^{2}+x-12\right)+3\left(5x-3\right)\right\}$

$=-2\left(2x-19-4{x}^{2}-2x+24+15x-9\right)$

$=-2\left(-4{x}^{2}+15x-4\right)$

$=8{x}^{2}-30x+8$　 ･･･（4）

$\left|\begin{array}{ccc}x& 2& 3\\ 1& 4& 5\\ 2& 2x& 6\end{array}\right|$

$=2×{\left(-1\right)}^{1+2}\left|\begin{array}{cc}1& 5\\ 2& 6\end{array}\right|+4×{\left(-1\right)}^{2+2}\left|\begin{array}{cc}x& 3\\ 2& 6\end{array}\right|+2x×{\left(-1\right)}^{3+2}\left|\begin{array}{cc}x& 3\\ 1& 5\end{array}\right|$

$=-2\left|\begin{array}{cc}1& 5\\ 2& 6\end{array}\right|+4\left|\begin{array}{cc}x& 3\\ 2& 6\end{array}\right|-2x\left|\begin{array}{cc}x& 3\\ 1& 5\end{array}\right|$

$=-2\left(6-10\right)+4\left(6x-6\right)-2x\left(5x-3\right)$

$=-2\left(-4\right)+4\left(6x-6\right)-2x\left(5x-3\right)$

$=8+24x-24-{10x}^{2}+6x$

$=-{10x}^{2}+30x-16$　 ･･･（5）

（1）に（2）〜（5）をそれぞれ代入する．

$-\left|\begin{array}{ccc}1& 4& 5\\ 2& 2x& 6\\ 4& 6& 2x+1\end{array}\right|$ $+\left(x+1\right)\left|\begin{array}{ccc}x& 2& 3\\ 2& 2x& 6\\ 4& 6& 2x+1\end{array}\right|$ $-3\left|\begin{array}{ccc}x& 2& 3\\ 1& 4& 5\\ 4& 6& 2x+1\end{array}\right|$ $+5\left|\begin{array}{ccc}x& 2& 3\\ 1& 4& 5\\ 2& 2x& 6\end{array}\right|$

$=-\left(4{x}^{2}-54x+112\right)$$+\left(x+1\right)×\left(4{x}^{3}+2{x}^{2}-68x+80\right)$$-3\left(8{x}^{2}-30x+8\right)$$+5\left(-10{x}^{2}+30x-16\right)$

$=-4{x}^{2}+54x-112+4{x}^{4}$$+6{x}^{3}-66{x}^{2}+12x+80$$-24{x}^{2}+90x-24$$-50{x}^{2}+150x-80$

$=4{x}^{4}+6{x}^{3}-144{x}^{4}+306x-136$

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