3

$\displaystyle \left(\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right.$ $\displaystyle \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right)$

�հݤU�C��@�ӿﶵ��x�}��n�N�� A �i��W�z�B�J�H

(1) $\left(\vphantom{ \begin{array}{cc} -4 & 0 \\ 0 & 0 \\ \end{array}%% }\right.$ $\begin{array}{cc} -4 & 0 \\ 0 & 0 \\ \end{array}$ $\left.\vphantom{ \begin{array}{cc} -4 & 0 \\ 0 & 0 \\ \end{array}%% }\right)$ $\left(\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right.$ $\begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}$ $\left.\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right)$ (2) $\left(\vphantom{ \begin{array}{cc} 0 & 0 \\ -4 & 0 \\ \end{array}%% }\right.$ $\begin{array}{cc} 0 & 0 \\ -4 & 0 \\ \end{array}$ $\left.\vphantom{ \begin{array}{cc} 0 & 0 \\ -4 & 0 \\ \end{array}%% }\right)$ $\left(\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right.$ $\begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}$ $\left.\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right)$

(3) $\left(\vphantom{ \begin{array}{cc} 1 & 0 \\ -4 & 0 \\ \end{array}%% }\right.$ $\begin{array}{cc} 1 & 0 \\ -4 & 0 \\ \end{array}$ $\left.\vphantom{ \begin{array}{cc} 1 & 0 \\ -4 & 0 \\ \end{array}%% }\right)$ $\left(\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right.$ $\begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}$ $\left.\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right)$ (4) $\left(\vphantom{ \begin{array}{cc} 0 & 0 \\ -4 & 1 \\ \end{array}%% }\right.$ $\begin{array}{cc} 0 & 0 \\ -4 & 1 \\ \end{array}$ $\left.\vphantom{ \begin{array}{cc} 0 & 0 \\ -4 & 1 \\ \end{array}%% }\right)$ $\left(\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right.$ $\begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}$ $\left.\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right)$

(5) $\left(\vphantom{ \begin{array}{cc} 1 & 0 \\ -4 & 1 \\ \end{array}%% }\right.$ $\begin{array}{cc} 1 & 0 \\ -4 & 1 \\ \end{array}$ $\left.\vphantom{ \begin{array}{cc} 1 & 0 \\ -4 & 1 \\ \end{array}%% }\right)$ $\left(\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right.$ $\begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}$ $\left.\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right)$

��:��D�D�N�j�ǽu�ʥN�Ƥ��򥻦C�B��i�H��x�}��k�o�Өƹ�J�D. ��L�@�뤣��D�o�өʽ誺�P�Ǥ��i�B�z��D�A�u�n�Q�ίx�}��k�@�@��Y�i. �٦n�o�ǯx�}��k��, �ҥH��D�o�өʽ誺�P�ǨèS�Y�h��.

�򥻦C�B��i�H��x�}��k�o�Өƹ��ܦn��, �\�h��ϯx�}��ʽ賣�i�H�ѥ��o��. �ҥH�ڭ̧Ʊ�Q�γo��D�بӻ��@�U�o�Өƹ�. ��ڭ̥O U = $\left(\vphantom{ \begin{array}{cc} a & b \\ c & d \\ \end{array}%% }\right.$ $\begin{array}{cc} a & b \\ c & d \\ \end{array}$ $\left.\vphantom{ \begin{array}{cc} a & b \\ c & d \\ \end{array}%% }\right)$ �䤤 (a b) �� U ��Ĥ@�C, �� (c d ) �� U ��ĤG�C. U �O�@�� 2×2 �x�}, ��D�ؤ�� A �O�@�� 2×3 �x�}. �ڭ̳��D U �i�H��b A ��, �B�� (�Y UA) ��O 2×3 ��x�} (�`�N U ��୼�b A��k��). �ڭ̨Ӭݬ� U ��b A ��|��?

�̯x�}��k�w�q, �b�D UA ��˪��x�}��, U ��Ĥ@�C (�O�@�� 1×2 �x�}) ��H A ��|�O UA �o�ӯx�}��Ĥ@�C, �� U ��ĤG�C��H A ��|�O UA �o�ӯx�}��ĤG�C. �ҥH�ڭ̥u�n��D U ��C�@�C��H A �ᬰ��, �N�i�D�o UA. ��ڭ̨Ӭ� U ��Ĥ@�C (a b) ��H A ��|��. �Y,

$\displaystyle \left(\vphantom{ \begin{array}{cc} a & b \\ \end{array}%% }\right.$ $\displaystyle \begin{array}{cc} a & b \\ \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{cc} a & b \\ \end{array}%% }\right)$ $\displaystyle \left(\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right.$ $\displaystyle \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ 4 & -5 & -1 \\ \end{array}%% }\right)$	=	$\displaystyle \left(\vphantom{ \begin{array}{ccc} a\cdot 1+b\cdot 4 & a\cdot 2+b\cdot(-5) & a\cdot 3+b\cdot(-1) \\ \end{array}%% }\right.$ $\displaystyle \begin{array}{ccc} a\cdot 1+b\cdot 4 & a\cdot 2+b\cdot(-5) & a\cdot 3+b\cdot(-1) \\ \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{ccc} a\cdot 1+b\cdot 4 & a\cdot 2+b\cdot(-5) & a\cdot 3+b\cdot(-1) \\ \end{array}%% }\right)$
	=	a $\displaystyle \left(\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ \end{array}%% }\right.$ $\displaystyle \begin{array}{ccc} 1 & 2 & 3 \\ \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{ccc} 1 & 2 & 3 \\ \end{array}%% }\right)$ + b $\displaystyle \left(\vphantom{ \begin{array}{ccc} 4 & -5 & -1 \\ \end{array}%% }\right.$ $\displaystyle \begin{array}{ccc} 4 & -5 & -1 \\ \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{ccc} 4 & -5 & -1 \\ \end{array}%% }\right)$ .

�{�b�^��ڭ̪��D. ��D�N�ڭ̧Ʊ�� U �ϱo UA ��Ĥ@�C��O A ��Ĥ@�C�ҥH�ѤW��G�ڭ̪��Y U ��Ĥ@�C�� (a b), �h�i�O a = 1 �B b = 0 �o�� UA ��Ĥ@�C�� (1 2 3). �P�z�Y UA ��ĤG�C�� A ��ĤG�C��h 4 �� A ��Ĥ@�C, (�Y -4(1 2 3) + (4 - 5 - 1)), �i�O U ��ĤG�C�� (- 4 1). �]�N�O��Y�O U = $\left(\vphantom{ \begin{array}{cc} 1 & 0 \\ -4 & 1 \\ \end{array}%% }\right.$ $\begin{array}{cc} 1 & 0 \\ -4 & 1 \\ \end{array}$ $\left.\vphantom{ \begin{array}{cc} 1 & 0 \\ -4 & 1 \\ \end{array}%% }\right)$ , �h UA �N�O�N A �Q��D�ةҭn�D��򥻦C�B��ұo��x�}. �ܩ��L��ذ򥻦C�B��, �Y�Y��C��H�άY�@�C��W�`��, �۫H�j�a�i�H�p�k�w�s��W��D�p��o��.

�@��ӻ�, �� A �O�@�� m×n ��x�}. �Y�n��@�� m×m ��x�} U �ϱo UA �N�O�N A ��Y�Ӱ򥻦C�B��ұo��x�}, ��ɧڭ̥u�n�N m×m ��x�}��ҭn�D��򥻦C�B��, �o�˩ұo��x�}�N�O�ҭn�D�� U �F. �Ҧp��D��ڭ̥i�N 2×2 ��x�} $\left(\vphantom{ \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array}%% }\right.$ $\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array}$ $\left.\vphantom{ \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array}%% }\right)$ ��ӳW�w��򥻦C�B�� (�Y�Ĥ@�C��, �ñN�ĤG�C��h�Ĥ@�C ��|��s��ĤG�C) �� $\left(\vphantom{ \begin{array}{cc} 1 & 0 \\ -4 & 1 \\ \end{array}%% }\right.$ $\begin{array}{cc} 1 & 0 \\ -4 & 1 \\ \end{array}$ $\left.\vphantom{ \begin{array}{cc} 1 & 0 \\ -4 & 1 \\ \end{array}%% }\right)$ �Y��ҨD.