2.
設\(A\)為三階可逆方陣且滿足\(A\left[\matrix{10\cr 10\cr 30}\right]=\left[\matrix{1\cr 0\cr 0}\right]\),\(A\left[\matrix{30\cr 40\cr 50}\right]=\left[\matrix{0\cr 20\cr 0}\right]\),\(A\left[\matrix{-20\cr -30\cr -40}\right]=\left[\matrix{0\cr 0\cr 50}\right]\),若\(AX=\left[\matrix{2\cr -40\cr 200}\right]\)且\(X=\left[\matrix{a\cr b\cr c}\right]\),則\((a,b,c)=\)?(不用三階反方陣的公式解題)
[解答]
\(\displaystyle (2,-40,200)=2(1,0,0)-2(0,20,0)+4(0,0,50)\)
所求\(\displaystyle X= 2(10,10,30)-2(30,40,50)+4(-20,-30,-40)=(-120,-180,-200)\)