線形代数-1-基礎問題-2

どうも、いもけんぴぃです。

今回は設問数と計算量が多いので１題のみです！

$A$ $= \begin{array}{r} (\begin{array}{cc} 1 & 2 & - 1 \\ 3 & 1 & 0 \end{array}) \end{array}$ , $B$ $= \begin{array}{r} (\begin{array}{cc} 2 & 0 & 1 \\ - 1 & 1 & 0 \\ 0 & 1 & 3 \end{array}) \end{array}$ , $a$ $= \begin{array}{r} (\begin{array}{cc} 1 \\ - 2 \\ 1 \end{array}) \end{array}$ , $b$ $= \begin{array}{r} (\begin{array}{cc} 3 & 2 \end{array}) \end{array}$

のとき,次の計算をせよ.

$(1)$ $^{t} A A$ $(2)$ $A^{t} A$ $(3)$ $B^{2}$ $(4)$ $A B$ $(5)$ $^{t} a B a$ $(6)$ $a b$

<解説>
今回は計算を頑張る回です.
面倒かもしれませんが耐えましょう.

$(1)$ $^{t} A A$ $=$ $\begin{array}{r} (\begin{array}{cc} 1 & 3 \\ 2 & 1 \\ - 1 & 0 \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} 1 & 2 & - 1 \\ 3 & 1 & 0 \end{array}) \end{array}$

$= \begin{array}{r} (\begin{array}{cc} 1 \times 1 + 3 \times 3 & 1 \times 2 + 3 \times 1 & 1 \times (- 1) + 3 \times 0 \\ 2 \times 1 + 1 \times 3 & 2 \times 2 + 1 \times 1 & 2 \times (- 1) + 1 \times 0 \\ - 1 \times 1 & - 1 \times 2 & - 1 \times (- 1) \end{array}) \end{array}$

$= \begin{array}{r} (\begin{array}{cc} 10 & 5 & - 1 \\ 5 & 5 & - 2 \\ - 1 & - 2 & 1 \end{array}) \end{array}$

$(2)$ $A^{t} A$ $=$ $\begin{array}{r} (\begin{array}{cc} 1 & 2 & - 1 \\ 3 & 1 & 0 \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} 1 & 3 \\ 2 & 1 \\ - 1 & 0 \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} 1 \times 1 + 2 \times 2 + (- 1) \times (- 1) & 1 \times 3 + 2 \times 1 + (- 1) \times 0 \\ 3 \times 1 + 1 \times 2 + 0 \times (- 1) & 3 \times 3 + 1 \times 1 + 0 \times 0 \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} 6 & 5 \\ 5 & 10 \end{array}) \end{array}$

$(3)$ $B^{2}$ $=$ $\begin{array}{r} (\begin{array}{cc} 2 & 0 & 1 \\ - 1 & 1 & 0 \\ 0 & 1 & 3 \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} 2 & 0 & 1 \\ - 1 & 1 & 0 \\ 0 & 1 & 3 \end{array}) \end{array}$

$= \begin{array}{r} (\begin{array}{cc} 2 \times 2 + 0 \times (- 1) + 1 \times 0 & 2 \times 0 + 0 \times 1 + 1 \times 1 & 2 \times 1 + 0 \times 0 + 1 \times 3 \\ - 1 \times 2 + 1 \times (- 1) + 0 \times 1 & - 1 \times 0 + 1 \times 1 + 0 \times 1 & - 1 \times 1 + 1 \times 0 + 0 \times 3 \\ 0 \times 2 + 1 \times (- 1) + 3 \times 0 & 0 \times 0 + 1 \times 1 + 3 \times 1 & 0 \times 1 + 1 \times 0 + 3 \times 3 \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} 4 & 1 & 5 \\ - 3 & 1 & - 1 \\ - 1 & 4 & 9 \end{array}) \end{array}$

$(4)$ $A B$ $=$ $\begin{array}{r} (\begin{array}{cc} 1 & 2 & - 1 \\ 3 & 1 & 0 \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} 2 & 0 & 1 \\ - 1 & 1 & 0 \\ 0 & 1 & 3 \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} 1 \times 2 + 2 \times (- 1) + - 1 \times 0 & 1 \times 0 + 2 \times 1 + - 1 \times 1 & 1 \times 1 + 2 \times 0 + - 1 \times 3 \\ 3 \times 2 + 1 \times (- 1) + 0 \times 0 & 3 \times 0 + 1 \times 1 + 0 \times 1 & 3 \times 1 + 1 \times 0 + 0 \times 3 \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} 0 & 1 & - 2 \\ 5 & 1 & 3 \end{array}) \end{array}$

$(5)$ $^{t} a B a$ $=$ $\begin{array}{r} (\begin{array}{cc} 1 & - 2 & 1 \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} 2 & 0 & 1 \\ - 1 & 1 & 0 \\ 0 & 1 & 3 \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} 1 \\ - 2 \\ 1 \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} 1 \times 2 + (- 2) \times (- 1) + 1 \times 0 & 1 \times 0 + (- 2) \times 1 + 1 \times 1 & 1 \times 1 + (- 2) \times 0 + 1 \times 3 \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} 1 \\ - 2 \\ 1 \end{array}) \end{array}$

$=$ $\begin{array}{r} (\begin{array}{cc} 4 & 3 & 4 \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} 1 \\ - 2 \\ 1 \end{array}) \end{array}$

$=$ $4 \times 1 + 3 \times (- 2) + 4 \times 1$

$=$ $10$

$(6)$ $a b$ $=$ $\begin{array}{r} (\begin{array}{cc} 1 \\ - 2 \\ 1 \end{array}) \end{array}$ $\begin{array}{r} (\begin{array}{cc} 3 & 2 \end{array}) \end{array}$